Constant-Rank Codes and Their Connection to ... - Semantic Scholar
Constant-Rank Codes and Their Connection to Constant-Dimension Codes Maximilien Gadouleau and Zhiyuan Yan Department of Electrical and Computer Engineering
arXiv:0803.2262v5 [cs.IT] 28 Jul 2008
Lehigh University, PA 18015, USA E-mails: {magc, yan}@lehigh.edu
Abstract Constant-dimension codes have recently received attention due to their significance to error control in noncoherent random network coding. What the maximal cardinality of any constant-dimension code with finite dimension and minimum distance is and how to construct the optimal constant-dimension code (or codes) that achieves the maximal cardinality both remain open research problems. In this paper, we introduce a new approach to solving these two problems. We first establish a connection between constantrank codes and constant-dimension codes. Via this connection, we show that optimal constant-dimension codes correspond to optimal constant-rank codes over sufficiently large extension fields. As such, the two aforementioned problems are equivalent to determining the maximum cardinality of constant-rank codes and to constructing optimal constant-rank codes, respectively. To this end, we derive bounds on the maximum cardinality of a constant-rank code with a given minimum rank distance, propose explicit constructions of optimal or asymptotically optimal constant-rank codes, and establish asymptotic bounds on the maximum rate of a constant-rank code.
I. I NTRODUCTION While random network coding [1]–[3] has proved to be a powerful tool for disseminating information in networks, it is highly susceptible to errors caused by various sources such as noise, malicious or malfunctioning nodes, or insufficient min-cut. If received packets are linearly combined at random to deduce the transmitted message, even a single error in one erroneous packet could render the entire transmission useless. Thus, error control for random network coding is critical and has received growing attention recently. Error control schemes proposed for random network coding assume two types of transmission models: some [4]–[8] depend on and take advantage of the underlying network topology
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or the particular linear network coding operations performed at various network nodes; others [9], [10] assume that the transmitter and receiver have no knowledge of such channel transfer characteristics. The contrast is similar to that between coherent and noncoherent communication systems. Error control for noncoherent random network coding was first considered in [9]1 . Motivated by the property that random network coding is vector-space preserving, an operator channel that captures the essence of the noncoherent transmission model was defined in [9]. Similar to codes defined in complex Grassmannians for noncoherent multiple-antenna channels, codes defined in Grassmannians over a finite field [12], [13] and used with the subspace distance (cf. [9, (3)]) play a significant role in error control for noncoherent random network coding; Under the subspace distance, the weight of a subspace is simply its dimension; thus, we refer to these codes as constant-dimension codes (CDCs) henceforth. The standard advocated approach to random network coding (see, e.g., [2]) involves transmission of packet headers used to record the particular linear combination of the components of the message present in each received packet. From coding theoretic perspective, the set of subspaces generated by the standard approach may be viewed as a suboptimal CDC with minimum subspace distance 2 in the Grassmannian, because the Grassmannian contains more spaces with minimum subspace distance 2 than those obtained by the standard approach [9]. Hence, studying random network coding from coding theoretic perspective results in better error control schemes. General studies of subspace metric codes (also referred to as codes in projective space or projective geometry) started only recently (see, for example, [14], [15]). On the other hand, there is a steady stream of works that focuses on codes in the Grassmannian. For example, Delsarte [12] proved that the Grassmannian endowed with the subspace distance forms an association scheme, and derived its parameters. The nonexistence of perfect codes in the Grassmannian was proved in [13], [16]. In [17], it was shown that Steiner structures yield diameter-perfect codes in the Grassmannian; properties and constructions of these structures were studied in [18]; in [19], it was shown that Steiner structures result in optimal CDCs. Related work on certain intersecting families and on byte-correcting codes can be found in [20] and [21], respectively. An application of codes in the Grassmanian to linear authentication schemes was considered in [22]. In [9], a Singleton bound for CDCs and a family of codes that are nearly Singleton-bound achieving are proposed, and a recursive construction of CDCs which outperform the codes in [9] was given in [23]. Despite the asymptotic optimality of the Singleton bound and the codes proposed in [9], both are not optimal in finite cases: upper bounds tighter than the Singleton bound 1
A related work [11] considers security issues in noncoherent random network coding.
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exist and can be achieved in some special cases [19]. It is yet to be determined the maximal cardinality of a CDC with finite dimension and minimum distance, and it is not clear how to construct the optimal code (or codes) that achieves the maximal cardinality. In this paper, we introduce a novel approach to solving the two aforementioned problems. Namely, we aim to solve these problems via rank metric codes, in particular, constant-rank codes (CRCs), which are the counterparts in rank metric codes of constant Hamming weight codes. There are several reasons for our approach. First, it is difficult to answer the two questions based on CDCs directly since the projective space lacks a natural group structure [10]. Second, the rank metric is similar to the Hamming metric and hence familiar results from the Hamming space can be readily adapted. Furthermore, there are extensive works on rank metric codes in the literature. Finally, the rank metric has been shown relevant to error control for both noncoherent [10] and coherent [24] random network coding. Based on our approach, this paper makes two main contributions. Our first main contribution is that we establish a connection between CRCs and CDCs. Via this connection, we show that optimal CDCs correspond to optimal CRCs over sufficiently large extension fields. This connection converts the aforementioned open research problems about CDCs into research problems about CRCs, thereby allowing us to use rich results in rank metric codes to tackle such problems. Since constant-rank codes have received little attention in the literature, our second main contribution is our investigation of the properties of CRCs. In particular, we derive upper and lower bounds on the maximum cardinality of a CRC, propose explicit constructions of optimal or asymptotically optimal CRCs, and establish asymptotic bounds on the maximum rate of CRCs. The rest of the paper is organized as follows. Section II reviews some necessary background. In Section III, we determine the connection between optimal CRCs and optimal CDCs. In Section IV, we study the maximum cardinality of CRCs, and we present our results on the asymptotic behavior of the maximum rate of a CRC. II. P RELIMINARIES A. Rank metric codes and elementary linear subspaces Error correction codes with the rank metric [25]–[27] have been receiving steady attention in the literature due to their applications in storage systems [27], public-key cryptosystems [28], space-time coding [29], and network coding [9], [10]. Below we review some important properties of rank metric codes established in [25]–[27].
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Consider a vector x of length n over GF(q m ). The field GF(q m ) may be viewed as an m-dimensional vector space over GF(q). The rank weight of x, denoted as rk(x), is defined to be the maximum number of coordinates of x that are linearly independent over GF(q) [26]. For any basis Bm of GF(q m ) over GF(q), each coordinate of x can be expanded to an m-dimensional column vector over GF(q) with
respect to Bm . Thus the rank weight of x is also given by the rank of the m × n matrix over GF(q) obtained by expanding all the coordinates of x [26]. We shall assume that the expansions are with respect to a given basis Bm of GF(q m ) over GF(q) henceforth. def
For all x, y ∈ GF(q m )n , it is easily verified that dR (x, y) = rk(x − y) is a metric over GF(q m )n , referred to as the rank metric henceforth [26]. The minimum rank distance and rank diameter of a code C , denoted as dR and DR , respectively, are simply the minimum and maximum rank distance over all
possible pairs of distinct codewords. It is shown in [25]–[27] that the minimum rank distance of a block code of length n and cardinality M over GF(q m ) satisfies dR ≤ n − logqm M + 1. In this paper, we refer to this bound as the Singleton
bound for rank metric codes and codes that attain the equality as maximum rank distance (MRD) codes. We refer to the subclass of linear MRD codes introduced in [30] as generalized Gabidulin codes. We denote the number of vectors of rank r (0 ≤ r ≤ min{m, n}) in GF(q m )n as Nr (q m , n) = n def n def def Qr−1 m i i=0 (q − q ), and r = α(n, r)/α(r, r) for r ≥ 1. r α(m, r) [26], where α(m, 0) = 1, α(m, r) = The nr term is often referred to as a Gaussian polynomial [31]. The volume of a ball with rank radius P r in GF(q m )n is denoted as Vr (q m , n) = ri=0 Ni (q m , n). We denote the intersection of two spheres in GF(q m )n of radii r and s, with distance between their centers d as J(q m , n, r, s, d). We will omit the
dependence of the quantities defined above on q m and n when there is no ambiguity about the vector space considered. By [32, Theorem 3.6], we obtain n X mn Nl Pr (l)Ps (l)Pd (l), q Nd J(r, s, d) =
(1)
l=0
where Pj (l) is a q -Krawtchouk polynomial [33], [34]: n X n−i j−i im+(j−i)(j−i−1)/2 n − l (−1) q Pj (l) = . n−j l
(2)
i=0
For all q , 1 ≤ d ≤ r ≤ n ≤ m, the number of codewords of rank r in an (n, n − d + 1, d) linear MRD code over GF(q m ) is given by [26] X r def n r−j r (r−j)(r−j−1)/2 m (−1) M (q , n, d, r) = q m(j−d+1) − 1 . q r j
(3)
j=d
An elementary linear subspace (ELS) [35] is defined to be a linear subspace V ⊆ GF(q m )n for which there exists a basis of vectors in GF(q)n . We denote the set of all ELSs of GF(q m )n with dimension
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v as Ev (q m , n) and the set of all ELSs as E(q m , n). It can be easily shown that |Ev (q m , n)| =
n v
for
all m. An ELS has properties similar to those for a set of coordinates [35]. In particular, any vector belonging to an ELS with dimension r has rank no more than r ; conversely, any vector x ∈ GF(q m )n with rank r belongs to a unique ELS in Er (q m , n). B. Subspace distance and constant-dimension codes For two subspaces of GF(q)n , U and V , it is easily verified that def
dS (U, V) = dim(U + V) − dim(U ∩ V) = 2 dim(U + V) − dim(U) − dim(V)
(4)
is a metric over E(q, n), referred to as the subspace metric [9] henceforth. The subspace distance between U and V thus satisfies dS (U, V) = 2rk(XT Y T )−rk(X)−rk(Y), where X and Y are generator matrices
of U and V , respectively. The minimum subspace distance and subspace diameter of any subspace metric code, denoted as dS and DS , respectively, are the minimum and maximum subspace distance over all possible pairs of distinct subspaces. A constant-dimension code Ω of length n and constant-dimension r over GF(q) is defined to be a nonempty subset of Er (q, n) [9]. When Ω has minimum subspace distance dS , we refer to Ω as an (n, dS , r) CDC over GF(q) and we denote the maximum cardinality of all (n, dS , r) CDCs over GF(q)
as AS (q, n, dS , r). Since AS (q, n, dS , r) = AS (q, n, dS , n − r) [19], only the case where 2r ≤ n needs to be considered. By (4), it is easily verified that the subspace distance between any two subspaces of the same dimension is even; thus, the minimum distance of any CDC is even. Hence, we shall consider AS (q, n, 2d, r) for 2 ≤ d ≤ r henceforth since AS (q, n, 2, r) = nr and AS (q, n, 2d, r) = 1 for d > r .
Upper and lower bounds on AS (q, n, 2d, r) were derived in [9], [14], [15], [19], [22]. In particular, for all q , 2r ≤ n, and 2 ≤ d ≤ r , it was shown in [9], [22] that q (n−r)(r−d+1) ≤ AS (q, n, 2d, r) ≤
α(n, r − d + 1) . α(r, r − d + 1)
(5)
Note that the lower bound is both constructive and asymptotically tight [9], and we use this bound in our derivation of asymptotic rate of CRCs. C. Preliminary graph-theoretic results We review some results in graph theory given in [36]. We denote the set of vertices in a graph G as V (G), and the adjacency between any two vertices u and v as u ∼ v .
Definition 1: Let G and H be two graphs. A mapping f from V (G) to V (H) is a homomorphism if for all u, v ∈ V (G), f (u) ∼ f (v) if u ∼ v .
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Definition 2: Let G be a graph and φ a bijection from V (G) to itself. φ is called an automorphism of G if for all u, v ∈ V (G), φ(u) ∼ φ(v) if and only if u ∼ v . Definition 3: The graph G is vertex transitive if for all u, v ∈ V (G), there exists an automorphism φ of G such that φ(u) = v . An independent set of a graph G is a subset of V (G) with no adjacent vertices. The independence number α(G) of G is the maximum cardinality of an independent set of G. If H is a vertex transitive graph and if there is a homomorphism from G to H , then [36], [37] α(G) ≥ α(H)
III. C ONNECTION
|G| . |H|
(6)
BETWEEN CONSTANT- DIMENSION CODES AND CONSTANT- RANK CODES
In this section, we first establish some connections between the rank metric and the subspace metric. We then define constant-rank codes and we show how optimal constant-rank codes can be used to construct optimal CDCs. For x ∈ GF(q m )n and a basis Bm of GF(q m ) over GF(q), let us consider X ∈ GF(q)m×n , the expansion of x with respect to Bm . Let us denote the row span and the column span of X over GF(q) as R(x; Bm ) and C(x; Bm ), respectively. Clearly, C(x; Bm ) ∈ Er (q, m) and R(x; Bm ) ∈ Er (q, n), where r = rk(x). We remark that R(x; Bm ) does not depend on Bm since changing a basis simply results in
elementary row operations on X; on the other hand, C(x; Bm ) depends on Bm , even the order of the elements in Bm , although the dimension of C(x; Bm ) is independent of Bm . We shall henceforth assume that the basis and the order of the basis elements are fixed and simply use the notations R(x) and C(x). def
The notations introduced above are naturally extended to codes as follows: for C ⊆ GF(q m )n , C(C) = def
{C(c) : c ∈ C} and R(C) = {R(c) : c ∈ C}.
Lemma 1: For U ∈ Er (q, m), V ∈ Er (q, n), and x ∈ GF(q m )n with rank r , C(x) = U and R(x) = V if and only if X = GT H, where G ∈ GF(q)r×m is a generator matrix of U and H ∈ GF(q)r×n is a generator matrix of V . The proof of Lemma 1 is straightforward and hence omitted. We remark that X = GT H is referred to as a rank factorization [38]. Alternatively, x = bH, where the expansion (with respect to a basis of GF(q m ) over GF(q)) of b ∈ GF(q m )r is given by GT . We now derive a relation between the rank
distance between two vectors and the subspace distances between their respective row and column spans. Theorem 1: For all x, y ∈ GF(q m )n , let us denote dS (C(x), C(y)) and dS (R(x), R(y)) as dC and dR , respectively. Without loss of generality, assume dC ≥ dR , then 1 1 max{dC + |rk(x) − rk(y)|, 2dC + dR − rk(x) − rk(y)} ≤ dR (x, y) ≤ [dR + rk(x) + rk(y)] . (7) 2 2
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Proof: We have R(x − y) ⊆ R(x) + R(y) and hence dR (x, y) = dim R(x − y) ≤ dim(R(x) + R(y)) = 12 (dR + rk(x) + rk(y)) by (4).
Let x = bH and y = b′ H′ so that b ∈ GF(q m )rk(x) and b′ ∈ GF(q m )rk(y) . By definition of the subspace distance, dim(C(x) ∩ C(y)) = 1 2
1 2
[rk(x) + rk(y) − dC ]. Therefore, we can select d1 =
[rk(x) − rk(y) + dC ] coordinates β0 , β1 , . . . , βd1 −1 of b linearly independent to b′ , and d2 = 12 [rk(y)−
rk(x) + dC ] coordinates β0′ , β1′ , . . . , βd′ 2 −1 of b′ linearly independent to b. By (4), it can be easily shown
that dC ≥ |rk(x) − rk(y)|. Let us select {γdC , γdC +1 , . . . , γm−1 } in GF(q m ) so that γ = {β0 , β1 , . . . , βd1 −1 , β0′ , β1′ , . . . , βd′ 2 −1 , γdC , γdC +1 , . . . , γm−1 }
constitutes a basis of GF(q m ) over GF(q). ¯T H ¯ ′T , where H ¯ and H ¯ ′ are the d1 Expanding x − y with respect to γ , we obtain rk(x − y) ≥ rk H
rows of H and the d2 rows of H′ corresponding to β0 , β1 , . . . , βd1 −1 and β0′ , β1′ , . . . , βd′ 2 −1 , respectively. ¯ rk(H ¯ ′ )} = max{d1 , d2 }. Second, since (HT H′T ) has rank ¯T H ¯ ′T ≥ max{rk(H), First, we have rk H 1 ¯ T ¯ ′T ≥ dC + 1 (dR − rk(x) − rk(y)). 2 (dR + rk(x) + rk(y)) by (4), rk H H 2
Definition 4: A constant-rank code of length n and constant-rank r over GF(q m ) is a nonempty subset
of GF(q m )n such that all elements have rank weight r . We denote a CRC with length n, minimum rank distance d, and constant-rank r as an (n, d, r) CRC over GF(q m ). Proposition 1 below shows how a CRC leads to two CDCs with their minimum subspace distance and subspace diameter related to the minimum rank distance and rank diameter of the CRC. Proposition 1: Let C be an (n, dR , r) CRC over GF(q m ) with rank diameter DR . Then R(C) ⊆ Er (q, n) is a CDC with minimum subspace distance dR ≥ 2(dR − r). Similarly, C(C) ⊆ Er (q, m) is a
CDC with minimum subspace distance dC ≥ 2(dR − r). If we denote the subspace diameters of R(C) and C(C) as DR and DC , respectively, then max{DR , DC } ≤ min{2DR , DR + r}. Proof: By definition, R(C) ⊆ Er (q, n) and C(C) ⊆ Er (q, m). Let x and y be two distinct codewords in C so that dS (R(x), R(y)) = dR . By Theorem 1, dR = dS (R(x), R(y)) ≥ 2(dR (x, y)−r) ≥ 2(dR − r), and similarly dC ≥ 2(dR − r).
Let x0 and y0 be two codewords in C such that dS (C(x0 ), C(y0 )) = DC . By Theorem 1, we have DC ≤ 2dR (x0 , y0 ) ≤ 2DR and 2DC ≤ 2dR (x0 , y0 ) + 2r − dS (R(x0 ), R(y0 )) ≤ 2DR + 2r . Let x1 and y1 be two codewords in C such that dS (R(x1 ), R(y1 )) = DR , then by Theorem 1, DR ≤ 2DR and 2DR ≤ dR (x0 , y0 ) + 2r − dS (C(x1 ), C(y1 )) ≤ 2DR + 2r .
When the minimum rank distance of a CRC is no less than its rank weight, Proposition 2 below shows how the CRC leads to two CDCs with the same cardinality, and the relations between their distances
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can be further strengthened. Proposition 2: If C is an (n, d + r, r) CRC over GF(q m ) (2 ≤ d ≤ r ) with rank diameter DR , then R(C) ⊆ Er (q, n) is a CDC with cardinality |C| and minimum subspace distance dR ≥ 2d. Similarly, C(C) ⊆ Er (q, m) is a CDC with cardinality |C| and minimum subspace distance dC ≥ 2d. We also have 1 2
max{2dC + dR − 2r, 2dR + dC − 2r} ≤ d + r . If we denote the subspace diameters of R(C) and C(C)
as DR and DC , respectively, then max{DR , DC } ≤ 2DR , min{DR , DC } ≥ 2DR − 2r , and 1 max{DC + 2dR , DR + 2dC , 2DC + dR , 2DR + dC } ≤ DR + r. 2 Proof: Let x and y be any two distinct codewords in C . By Theorem 1, dS (R(x), R(y)) ≥ 2dR (x, y) − 2r ≥ 2d > 0, and hence dR ≥ 2d and |R(C)| = |C|. Similarly, dS (C(x), C(y)) ≥ 2d > 0,
and thus dC ≥ 2d and |C(C)| = |C|. Furthermore, if dR (x, y) = d + r , then by Theorem 1, 2(d + r) ≥ 2dS (C(x), C(y)) + dS (R(x), R(y)) − 2r ≥ 2dC + dR − 2r . Similarly, we obtain 2(d + r) ≥ 2dR + dC − 2r .
We now prove the inequalities involving DC . Let x0 and y0 be two codewords in C such that DC = dS (C(x0 ), C(y0 )). By Theorem 1, we have DC ≤ 2dR (x0 , y0 ) ≤ 2DR , DC ≤ 2dR (x0 , y0 ) + 2r − 2dS (R(x0 ), R(y0 )) ≤ 2DR + 2r − 2dR , and 2DC ≤ 2dR (x0 , y0 ) + 2r − dS (R(x0 ), R(y0 )) ≤ 2DR + 2r − dR . Let x1 and y1 be two codewords in C such that dR (x1 , y1 ) = DR , then by Theorem 1, DR ≤
1 2
[dS (C(x1 ), C(y1 )) + 2r] ≤
1 2
[DC + 2r]. The other inequalities involving DR are obtained
similarly. Using Lemma 1, we can construct CRCs in GF(q m )n from a pair of CDCs in GF(q)n and GF(q)m , respectively. Proposition 3: Let Γ be an (m, dC , r) CDC over GF(q) and ∆ be an (n, dR , r) CDC over GF(q). Then there exists a CRC C with length n, constant-rank r , and cardinality min{|Γ|, |∆|} over GF(q m ) satisfying C(C) ⊆ Γ and R(C) ⊆ ∆. Furthermore, its minimum distance dR satisfies dR − 2r, 0} ≤ dR ≤
1 2
1 2
max{dC , dR } + 12 max{dC +
max{dC , dR } + r . If we denote the subspace diameters of ∆ and Γ as DR and
DC , respectively, then the rank diameter DR of C satisfies
1 2
min{DR , DC } ≤ DR ≤
1 2
min{DR , DC } + r
and min{DC + 2dR , DR + 2dC , 2DC + dR , 2DR + dC } ≤ 2DR + 2r. Proof: Denote the generator matrices of the component subspaces of Γ and ∆ as Gi and Hi , def
respectively. Define the code C formed by the codewords ci = bi Hi for 0 ≤ i ≤ min{|Γ|, |∆|} − 1, where the expansion of bi ∈ GF(q m )r is given by GTi . Then C(C) ⊆ Γ and R(C) ⊆ ∆ by Lemma 1 and the lower bound on dR follows Theorem 1. By construction, C(C) = Γ or R(C) = ∆. If C(C) = Γ, then let x and y be distinct codewords in C such that dS (C(x), C(y)) = dC . By Theorem 1, we obtain dR ≤ dR (x, y) ≤ 12 (dC + 2r). Similarly, if
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R(C) = ∆, we obtain dR ≤ 21 (dR + 2r). Combining these results, we obtain dR ≤
1 2
max{dR + 2r, dC +
2r}.
Let x and y be two codewords in C such that dR (x, y) = DR , then by Theorem 1, 2DR ≤ dS (R(x), R(y))+ 2r ≤ DS (R(C)) + 2r ≤ DR + 2r , and similarly 2DR ≤ DC + 2r . If C(C) = Γ, then let x′ and y′ be
distinct codewords in C such that dS (C(x), C(y)) = DC ; by Theorem 1, R(C) = ∆, we obtain 21 DR ≤ DR , and hence
1 2
1 2 DC
≤ dR (x′ , y′ ) ≤ DR . If
min {DC , DR } ≤ DR . By a similar argument, we obtain
min{DC + 2dR , DR + 2dC , 2DC + dR , 2DR + dC } ≤ 2DR + 2r.
Proposition 4: Let ∆ be an (n, 2d, r) CDC over GF(q) with subspace diameter DS . Then for any extension field GF(q m ) with m ≥ n, there exists a CRC C of length n and constant-rank r , over GF(q m ) such that R(C) = ∆ and |C| = |∆|. The minimum distance dR of C satisfies 2d ≤ dR ≤ d + r
and the rank diameter DR of C satisfies
1 2
max{DS , 3DS − 2r} ≤ DR ≤ 12 DS + r .
The proof of Proposition 4 follows Theorem 1, and is similar to that of Proposition 3. The proof is given in Appendix A. We denote the maximum cardinality of an (n, d, r) CRC over GF(q m ) as AR (q m , n, d, r). If C is an (n, d, r) CRC over GF(q m ), then the code obtained by transposing all the expansion matrices of codewords in C forms an (m, d, r) CRC over GF(q n ) with the same cardinality, and vice versa2 . Therefore AR (q m , n, d, r) = AR (q n , m, d, r), and henceforth in this paper we assume n ≤ m without loss of
generality. We further observe that AR (q m , n, d, r) is a non-decreasing function of m and n, and a nonincreasing function of d, and that AS (q, n, 2d, r) is a non-decreasing function of n and a non-increasing function of d. Proposition 5: For all q , 2 ≤ d ≤ r ≤ n ≤ m, and any 0 ≤ p ≤ r , min{AS (q, n, 2(d + 2p), r), AS (q, m, 2(r − p), r)} ≤ AR (q m , n, d + r, r) ≤ AS (q, n, 2d, r).
(8)
AR (q m , n, d + r, r) ≥ AS (q, n, d + r, r).
(9)
Also,
Proof: Using the monotone properties of AR (q m , n, dR , r) and AS (q, n, dS , r) above, the upper bound in (8) follows Proposition 2, while the lower bound in (8) follows Proposition 3 for dC = 2(r − p) and dR = 2(d + 2p). Finally, (9) follows Proposition 4 and the monotone properties of AR (q m , n, dR , r).
We remark that the lower bound in (8) is trivial for d + 2p > min{r, n − r} or r − p > min{r, m − r}. Therefore, the lower bound in (8) is nontrivial when max{0, 2r − m} ≤ p ≤ 2
However, the linearity of codes is not preserved through transposition.
1 2
min{r − d, n − r − d}.
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Combining the bounds in (8), we obtain that the cardinalities of optimal CRCs over sufficiently large fields are equal to the cardinalities of CDCs with related distances. Furthermore, we show that optimal CDCs can be constructed from such optimal CRCs. Theorem 2: For all q , 2r ≤ n ≤ m, and 1 ≤ d ≤ r , AR (q m , n, d + r, r) = AS (q, n, 2d, r) if either d = r or m ≥ m0 , where m0 = (n − r)(r − d + 1) + r + 1. Furthermore, if C is an (n, d + r, r) optimal
CRC over GF(q m ) for m ≥ m0 or d = r , then R(C) is an optimal (n, 2d, r) CDC over GF(q). Proof: First, the case where d = r directly follows (8) for p = 0. Second, if d < r and m ≥ m0 , by (5) we obtain AS (q, m, 2r, r) ≥ q m−r ≥ q m0 −r . Also, by [35, Lemma 1], we obtain q r(r−d+1)−1 < α(r, r − d + 1) ≤ q r(r−d+1) for all 2 ≤ d < r , and hence (5) yields AS (q, n, 2d, r) < q (n−r)(r−d+1)+1 = q m0 −r ≤ AS (q, m, 2r, r). Thus, when p = 0, the lower bound in (8) simplifies to AR (q m , n, d + r, r) ≥ AS (q, n, 2d, r). Combining with the upper bound in (8), we obtain AR (q m , n, d + r, r) = AS (q, n, 2d, r).
The second claim immediately follows Proposition 2. Theorem 2 implies that to determine AS (q, n, 2d, r) and to construct optimal CDCs, it is sufficient to determine AR (q m , n, d + r, r) and to construct optimal CRCs over an extension field sufficiently large. We observe that this implies that AR (q m , n, d + r, r) remains constant for all m ≥ m0 . When d = r , AR (q m , n, 2r, r) remains constant for m ≥ n. When d = 1, m0 = (n − r + 1)r + 1, but AR (q m , n, r + 1, r)
remains constant for m ≥ n, and this is shown in Section IV-C. IV. C ONSTANT- RANK
CODES
Having proved that optimal CRCs over sufficiently large extension fields lead to optimal CDCs, in this section we investigate the properties of CRCs. A. Graph-theoretic results for constant-rank codes We now define two families of graphs which are instrumental in our analysis of CRCs. Definition 5: The bilinear forms graph Rq (m, n, d) has as vertices all the vectors in GF(q m )n and two vertices x and y are adjacent if and only if dR (x, y) < d. The constant-rank graph Kq (m, n, d, r) is the subgraph of Rq (m, n, d) induced by the vectors in GF(q m )n with rank r . The orders of the bilinear forms and constant-rank graphs are thus given by |Rq (m, n, d)| = q mn and |Kq (m, n, d, r)| = Nr (q m , n). An independent set of Rq (m, n, d) corresponds to a code with minimum rank distance ≥ d. Due to the existence of MRD codes for all parameter values [39], we have α(Rq (m, n, d)) = q m(n−d+1) . Similarly, an independent set of Kq (m, n, d, r) corresponds to a CRC with minimum rank distance ≥ d, and hence α(Kq (m, n, d, r)) = AR (q m , n, d, r).
11
Lemma 2: The bilinear forms graph Rq (m, n, d) is vertex transitive for all q , m, n, and d. The constantrank graph Kq (m, m, d, m) is vertex transitive for all q , m, and d. Proof: For given u, v ∈ GF(q m )n , define φ(x) = x + v − u for all x ∈ GF(q m )n . It is easily shown that φ is a graph automorphism of Rq (m, n, d) satisfying φ(u) = v. By Definition 3, Rq (m, n, d) is hence vertex transitive. Let u, v ∈ GF(q m )m have rank m, and denote their expansions as U and V, respectively. For all x ∈ GF(q m )m with rank m, define φ(x) = y such that Y = XU−1 V, where X and Y are the
expansions of x and y, respectively. We have φ(u) = v, rk(φ(x)) = m, and for all x, z ∈ GF(q m )m , dR (φ(x), φ(z)) = rk(XU−1 V−ZU−1 V) = rk(X−Z) = dR (x, z). By Definition 2, φ is an automorphism
which maps u to v and hence Kq (m, m, d, m) is vertex transitive. It is worth noting that Kq (m, n, d, r) is not vertex transitive in general. B. Bounds We now derive bounds on the maximum cardinality of CRCs. We first remark that the bounds on AR (q m , n, d, r) derived in Section III can be used in this section. Also, since AR (q m , n, 1, r) = Nr (q m , n)
and AR (q m , n, d, r) = 1 for d > 2r , we shall assume 2 ≤ d ≤ 2r henceforth. We first derive the counterparts of the Gilbert and the Hamming bounds for CRCs in terms of intersections of spheres with rank radii. def
Proposition 6: For all q , 1 ≤ r, d ≤ n ≤ m, and t = ⌊ d−1 2 ⌋, Nr (q m , n) Nr (q m , n) ≤ AR (q m , n, d, r) ≤ Pt . (10) Pd−1 m m i=0 J(q , n, i, r, r) i=0 J(q , n, i, r, r) The proof is straightforward and hence omitted. The Hamming bound is generalized as follows. def
Proposition 7: For all q , 1 ≤ r, d, s ≤ n ≤ m, and t = ⌊ d−1 2 ⌋, Ns (q m , n) . (11) m i=0 J(q , n, i, s, r) be an (n, d, r) CRC over GF(q m ). For all 0 ≤ k ≤ K − 1 and
AR (q m , n, d, r) ≤ Pt
K−1 Proof: Let C = {ck }k=0
0 ≤ s ≤ n − 1, if we denote the set of vectors in GF(q m )n with rank s and distance ≤ t from ck Pt as Rk,s , then |Rk,s | = i=0 J(i, s, r) for all k . Clearly Rk,s ∩ Rl,s = ∅ for all k 6= l, and hence SK−1 Ns ≥ | k=0 Rk,s | = K|Rk,s |, which yields (11).
We now derive upper bounds on AR (q m , n, d, r). We begin by proving the counterpart in rank metric
codes of a well-known bound on constant-weight codes proved by Johnson in [40].
12
Proposition 8: For all q , 1 ≤ r, d < n ≤ m, AR (q m , m, d, m) ≤ q m−1 (q m − 1)AR (q m−1 , m − 1, d, m − 1) qn − 1 AR (q m , n − 1, d, r). q n−r − 1 The proof of Proposition 8 is given in Appendix B. AR (q m , n, d, r) ≤
(12) (13)
The Singleton bound for rank metric codes yields upper bounds on AR (q m , n, d, r). For any I ⊆ {0, 1, . . . , n}, let AR (q m , n, d, I) denote the maximum cardinality of a code of length n and minimum
rank distance d over GF(q m ) such that all codewords have rank weights belonging to I . def
Proposition 9: For all q , 1 ≤ r, d ≤ n ≤ m, define Pr = {i : 0 ≤ i ≤ n, |i − r| ≥ d} and def
Qr,a = Pr ∩ {a + kd : k ∈ Z} for all 0 ≤ a < d. Then AR (q m , n, d, r) ≤ q m(n−d+1) − AR (q m , n, d, Pr ) X X AR (q m , n, d, j) . M (q m , n, d, i), ≤ q m(n−d+1) − max
(14) (15)
j∈Qr,a
i∈Pr
Proof: For 0 ≤ j ≤ n, let Cj be an optimal (n, d, j) CRC and let C ⊆ GF(q m )n be a code with minimum rank distance d and whose codewords have rank weights belonging to Pr . Eq. (14) directly ˙ r , where ∪˙ denotes disjoint union. Let G be an (n, n − d + 1, d) follows the Singleton bound on C ∪C
linear MRD code over GF(q m ), and denote the subset of codewords with ranks belonging to Pr as G′ . def S Finally define C ′ = ˙ j∈Qr,a Cj . Both G′ and C ′ are codes in GF(q m )n with minimum rank distance d and whose codewords have rank weights belonging to Pr , hence AR (q m , n, Pr ) ≥ max{|G′ |, |C ′ |},
which leads to (15). We now determine the counterpart of the Singleton bound for CRCs. def
Proposition 10: For all 0 ≤ i ≤ min{d − 1, r}, Ji = {r − i, r − i + 1, . . . , min{n − i, r}}. Then AR (q m , n, d, r) ≤ AR (q m , n − i, d − i, Ji )
(16)
min{n−i,r}
≤
X
AR (q m , n − i, d − i, j).
(17)
j=r−i
Proof: Let C be an optimal (n, d, r) CRC over GF(q m ), and consider the code Ci obtained by puncturing i coordinates of the codewords in C . Since i ≤ r , the codewords of Ci all have ranks between r − i and min{n − i, r}. Also, since i < d, any two codewords have distinct puncturings, and we obtain |Ci | = |C| and dR (Ci ) ≥ d − i. Hence AR (q m , n, d, r) = |C| = |Ci | ≤ AR (q m , n − i, d − i, Ji ). (17)
directly follows a union bound on AR (q m , n − i, d − i, Ji ). We now combine the counterparts of the Johnson bound in (13) and of the Singleton bound in Proposition 10 in order to obtain an upper bound on AR (q m , n, d, r) for d ≤ r .
13
Proposition 11: For all q , 1 ≤ d ≤ r ≤ n ≤ m, AR (q m , n, d, r) ≤ Proof: Applying (13) n − r times successively, we obtain AR
n r
α(m, r − d + 1). ≤ nr AR (q m , r, d, r). For
(q m , n, d, r)
n = r and i = d − 1, Ji = {r − d + 1} and hence (16) yields AR (q m , r, d, r) ≤ AR (q m , r − d + 1, 1, r − d + 1) = Nr−d+1 (q m , r − d + 1) = α(m, r − d + 1). Thus AR (q m , n, d, r) ≤ nr α(m, r − d + 1).
We now derive the counterpart in rank metric codes of the Bassalygo-Elias bound [41]. We also tighten
the bound when d > r + 1. Proposition 12: For max{r, d} ≤ k ≤ n, 0 ≤ s ≤ k, k ≤ l ≤ m, and any code C ⊆ GF(q l )k with def
minimum rank distance d and rank weight distribution Ai = |{c ∈ C : rk(c) = i}|, Pn l m i=0 Ai J(q , k, s, r, i) . AR (q , n, d, r) ≥ max Ns (q l , k) s,{Ai },k,l
(18)
Furthermore, if r + 1 < d ≤ 2r , then m
AR (q , n, d, r) ≥
max
Pn
Ns (q l , k) − The proof of Proposition 12 is given in Appendix C. s,{Ai },k,l
Ai J(q l , k, s, r, i) . Pd−r−1 J(q l , k, s, t, i) i=0 Ai t=0
i=0 P n
(19)
Although the RHS of (18) and (19) can be maximized over {Ai }, it is difficult to do so since {Ai } is not available for most rank metric codes with the exception of linear MRD codes. Thus, we derive a bound using the rank weight distribution of linear MRD codes. Corollary 1: For all q , 1 ≤ r, d ≤ n ≤ m, AR (q m , n, d, r) ≥ Nr (q m , n)q m(−d+1) .
(20)
Proof: Applying (18) to an (n, n−d+1, d) MRD code over GF(q m ), we obtain Ns (q m , n)AR (q m , n, d, r) ≥ Pn Pn m m i=0 M (q , n, d, i)J(q , n, s, r, i). Summing for all 0 ≤ s ≤ n, we obtain (20) since s=0 J(s, r, i) =
Nr (q m , n).
We also give an alternate proof of (20) based on the results in Section IV-A. Since Kq (m, n, d, r) is a subgraph of Rq (m, n, d), the inclusion map is a trivial homomorphism from Kq (m, n, d, r) to Rq (m, n, d). By Lemma 2, Rq (m, n, d) is vertex transitive. We hence apply (6) to these graphs, which yields (20). The RHS of (18) and (19) decrease rapidly with increasing d, rendering the bounds trivial for d approaching 2r . We investigate below the tightness of the bound in Corollary 1. def
Proposition 13: For all q , 2 ≤ d ≤ r ≤ n ≤ m, let C(q m , n, d, r) = AR (q m , n, d, r)/[Nr (q m , n)q m(−d+1) ]. Then C(q m , n, d, r) ≤ C(q m , n, d, r)
C(q m , n, d, r) ≤
q−1 −1 q Kq .
Finally,
all 1 ≤ l ≤ n − 1 [35, Lemma 1], we obtain for 2l ≤ n [35, Lemma 1] yields (21).
It is worth noting that Kq above represents the fraction of invertible m × m matrices over GF(q) as m approaches infinity. Kq−1 decreases with q and satisfies 1 < Kq−1 ≤ K2−1 < 4. Thus the bound in Corollary 1 is tight up to a scalar when d ≤ r . We also remark that the bound in (21) is tighter than that in (22). However, these bounds are not constructive. Below we derive constructive bounds on AR (q m , n, d, r).
C. Constructive bounds We now give explicit constructions of good CRCs when d ≤ r , which in turn yield asymptotically tight lower bounds on AR (q m , n, d, r). Proposition 14: For all q , 2 ≤ d ≤ r ≤ n ≤ m, AR (q m , n, d, r) ≥ M (q m , n, d, r) >
n r
q m(r−d) .
Proof: The codewords of rank r in an (n, n − d + 1, d) linear MRD code over GF(q m ) form an (n, d, r) CRC. Thus, AR (q m , n, d, r) ≥ M (q m , n, d, r). We now prove the lower bound on M (q m , n, d, r). First, for d = r , M (q m , n, r, r) = nr (q m − 1) > nr . Second, suppose d < r . By (3), M (q m , n, d, r) can be expressed as M (q m , n, d, r) = n Pr r−j µ , where µ def (r−j)(r−j−1)/2 r (q m(j−d+1) − 1). It can be easily shown that µ > (−1) = q j j j j=d r j n µj−1 for d + 1 ≤ j ≤ r , and hence M (q m , n, d, r) ≥ r (µr − µr−1 ). Therefore, M (q m , n, d, r) ≥ n m(r−d) r m(r−d) n m(r−d+1) . (q − 1)] > [(q − 1) − r q 1 r Corollary 2: For all q , 1 ≤ r ≤ n ≤ m, AR (q m , n, r, r) = nr (q m − 1). Proof: By Proposition 11, AR (q m , n, r, r) ≤ nr (q m − 1), and by Proposition 14, AR (q m , n, r, r) ≥ M (q m , n, r, r) = nr (q m − 1).
By Corollary 2, the codewords of rank r in an (n, n − r + 1, r) linear MRD code are optimal. We
investigate below the tightness of the constructive lower bound in Proposition 14. def
Proposition 15: For all q , 1 ≤ d < r ≤ n ≤ m, let B(q m , n, d, r) = AR (q m , n, d, r)/M (q m , n, d, r).
15
Then for m ≥ 3, B(2m , m, m − 1, m) ≤ 2m−1 − 1 B(q m , m, m − 1, m) < B(q m , m, m − 2, m)
2 q−2 (q 2 − 1)(q − 1) (q 2 − 1)(q − 2) + 1
(q 3 − 1)(q 2 − 1)(q − 1) (q 3 − 1)(q 2 − 1)(q − 2) + q 3 − 2 q B(q m , n, d, r) < for r < m. q−1 The proof of Proposition 15 is given in Appendix D. B(q m , m, d, m)
r using generalized Gabidulin codes [30]. Let g ∈ GF(q m )n have rank n, and for 0 ≤ i ≤ m − 1, denote the vector in GF(q m )n obtained by elevating each coordinate of g to the q ai -th power as g[i] , where a and m are coprime. Let C be the (n, n − d + 1, d) generalized T T T T , and C ′ be the (n, d − r, n − d + Gabidulin code over GF(q m ) generated by g[0] g[1] · · · g[n−d] T T T T . We consider the r + 1) generalized Gabidulin code generated by g[n−d+1] g[n−d+2] · · · g[n−r] coset C + c′ , where c′ ∈ C ′ , and we denote the number of codewords of rank r in C + c′ as σr (c′ ). Lemma 3: For all d > r , there exists c′ ∈ C ′ such that σr (c′ ) ≥ nr q m(r−d+1) .
Proof: Any codeword c′ ∈ C ′ can be expressed as c′ = cn−d+1 g[n−d+1] + cn−d+2 g[n−d+2] + . . . +
cn−r g[n−r] , where ci ∈ GF(q m ) for n − d + 1 ≤ i ≤ n − r . If cn−r = 0, then (C + c′ ) ⊂ D , where D is T T T T . Therefore the (n, n − r, r + 1) generalized Gabidulin code generated by g[0] g[1] · · · g[n−r−1] σr (c′ ) = 0 if cn−r = 0.
S Denote the number of codewords of rank r in C ⊕ C ′ as τr . Since c′ ∈C ′ (C + c′ ) = C ⊕ C ′ , we have P τr = c′ ∈C ′ σr (c′ ). Also, C ⊕ C ′ forms an (n, n − r + 1, r) MRD code, and hence τr = M (q m , n, r, r) = n m(r−d+1) n m P ′ ′ ′ ′ . Then τr = c′ :cn−r 6=0 σr (c ) < r (q − 1). Suppose that for all c ∈ C , σr (c ) < r q n m n m r (q − 1), which contradicts τr = r (q − 1).
Although Lemma 3 proves the existence of a vector c′ for which the translate C+c′ has high cardinality,
it does not indicate how to choose c′ . For d = r + 1, it can be shown that all c′ ∈ C ′ satisfy the bound, and that they all lead to optimal codes. Corollary 3: If d = r + 1, then σr (c′ ) =
n r
for all c′ ∈ C ′ .
Proof: First, by Proposition 5, σr (c′ ) ≤ AR (q m , n, r + 1, r) ≤ AS (q, n, 2, r) = nr for all c′ ∈ C ′ . Suppose there exists c′ such that σr (c′ ) < nr . Then τr < nr (q m −1), which contradicts τr = nr (q m −1).
16
Proposition 16: For all q , 1 ≤ r < d ≤ n ≤ m, AR (q m , n, d, r) ≥ that satisfy this bound can be constructed from Lemma 3.
n r
q n(r−d+1) , and a class of codes
Proof: The codewords of rank r in a code considered in Lemma 3 form an (n, d, r) CRC over GF(q m ) with cardinality ≥ nr q m(r−d+1) . Therefore, AR (q m , n, d, r) ≥ nr q m(r−d+1) . The proof is concluded by noting that AR (q m , n, d, r) ≥ AR (q n , n, d, r) ≥ nr q n(r−d+1) . Corollary 4: For all q , 1 ≤ r < n ≤ m, AR (q m , n, r + 1, r) = nr = AS (q, n, 2, r). Proof: Combine Proposition 5 and Proposition 16. We note that nr is independent of m. We also remark that the lower bound in Proposition 16 is also
trivial for d approaching 2r . Since the proof is only partly constructive, computer search can be used to help find better results for small parameter values. D. Asymptotic results
In this section, we study the asymptotic behavior of CRCs using the following set of normalized n m,
r m,
d m.
By definition, 0 ≤ ρ, δ ≤ ν , and since we assume n ≤ m, h i def ν ≤ 1. We consider the asymptotic rate defined as aR (ν, δ, ρ) = limm→∞ sup logqm2 AR (q m , n, d, r) .
parameters: ν =
ρ=
and δ =
We now investigate how AR (q m , n, d, r) behaves as the parameters tend to infinity. Without loss of generality, we only consider the case where 0 ≤ δ ≤ min{ν, 2ρ}, since aR (ν, δ, ρ) = 0 for δ > 2ρ. Proposition 17: For 0 ≤ δ ≤ ρ, aR (ν, δ, ρ) = ρ(1 + ν − ρ) − δ.
(28)
For ρ ≤ δ, we have to distinguish three cases. First, for 2ρ ≤ ν , (1 − ρ)(ν − ρ) ν−ρ max (2ρ − δ), (2ρ − δ), ρ(2ν − ρ) − νδ ≤ aR (ν, δ, ρ) ≤ (ν − ρ)(2ρ − δ). (29) 1 + 2ν − 3ρ 2 Second, for ν ≤ 2ρ ≤ 1, ρ(1 − ρ) ρ max (ν − δ), (2ν − 2ρ − δ), ρ(2ν − ρ) − νδ ≤ aR (ν, δ, ρ) ≤ ρ(ν − δ). 1+ρ 2 Third, for 2ρ ≥ 1, o nρ ρ (2 − 4ρ + ν − δ), (2ν − 2ρ − δ), ρ(2ν − ρ) − νδ, 0 ≤ aR (ν, δ, ρ) ≤ ρ(ν − δ). max 3 2 The proof of Proposition 17 is given in Appendix E.
(30)
(31)
Proposition 17 indicates that the codewords of rank r in an (n, n − d + 1, d) linear MRD code (d ≤ r ) form an asymptotically optimal (n, d, r) CRC. In particular, the set of codewords with rank n constitutes
17
a CRC of rank n and asymptotic rate of ν − δ, which is equal to the asymptotic rate of an optimal rank metric code [42]. The bounds on aR (ν, δ, ρ) given in Proposition 17 are illustrated in Figures 1, 2, and 3 for ν = 3/4 and ρ = 1/4, ρ = 2/5, and ρ = 3/5, respectively. Note that these three parameters correspond to the three cases in (29), (30), and (31), respectively.
0.4 δ≤ρ upper lower
0.35 0.3
aR(ν,δ,ρ)
0.25 0.2 0.15 0.1 0.05 0
Fig. 1.
0
0.1
0.2
δ
0.3
0.4
0.5
Asymptotic bounds on the maximal rate of a CRC as a function of δ, with ν = 3/4 and ρ = 1/4.
In Figures 1, 2, and 3, we can split the range of δ into two regions: when δ ≤ ρ, the asymptotic rate of CRCs is determined due to the construction of good CRCs when d ≤ r ; when δ ≥ ρ, we only have bounds on the asymptotic rate of CRCs. We remark that in (29), the first two lower bounds are competing, thus forming a triangle-shaped region for aR (ν, δ, ρ). However, for (30) and (31), the shape of the possible region for aR (ν, δ, ρ) depends on both ν and ρ. Also, the lower bounds based on the connection between CDCs and CRCs (the first two lower bounds in the LHS of (29), (30), and (31)) are tighter for 2ρ ≤ ν and on the other hand become trivial for ρ approaching 1.
18
0.7 δ≤ρ upper lower
0.6
aR(ν,δ,ρ)
0.5
0.4
0.3
0.2
0.1
0
Fig. 2.
0
0.1
0.2
0.3
0.4 δ
0.5
0.6
0.7
0.8
Asymptotic bounds on the maximal rate of a CRC as a function of δ, with ν = 3/4 and ρ = 2/5.
0.7 δ≤ρ upper lower
0.6
aR(ν,δ,ρ)
0.5
0.4
0.3
0.2
0.1
0
Fig. 3.
0
0.1
0.2
0.3
0.4 δ
0.5
0.6
0.7
0.8
Asymptotic bounds on the maximal rate of a CRC as a function of δ, with ν = 3/4 and ρ = 3/5.
19
A PPENDIX A. Proof of Proposition 4 Proof: Let us first focus on GF(q n ) and consider a basis Bn of GF(q n ) over GF(q). Define the def
code C over GF(q n ) formed by the codewords ci = bi Hi for 0 ≤ i ≤ |∆| − 1, where the expansion of bi ∈ GF(q n )r is given by HTi . Let us consider ci = bi Hi and cj = bj Hj . Let βi,k0 , βi,k1 , . . . , βi,kd−1 be
coordinates of bi linearly independent to the coordinates of bj and βj,l0 , βj,l1 , . . . , βj,ld−1 be coordinates of bj linearly independent to the coordinates of bi . We thus define the basis γi,j = {βi,k0 , βi,k1 , . . . , βi,kd−1 , βj,l0 , βj,l1 , . . . , βj,ld−1 , γ2d , γ2d+1 , . . . , γn−1 }
of GF(q n ) over GF(q). Expanding ci − cj with respect to the basis γi,j , we obtain rk(ci − cj ) ≥ ¯j ¯ i denotes the d rows of Hi corresponding to βi,k , βi,k , . . . , βi,k ¯ T , where H ¯T −H and H rk H d−1 1 0 j i denotes the d rows of Hj corresponding to βj,l0 , βj,l1 , . . . , βj,ld−1 . Thus
¯T −H ¯ T = rk(βi,k , βi,k , . . . , βi,k , −βj,l , −βj,l , . . . , −βj,l ) = 2d. rk H i j d−1 1 0 d−1 1 0
Using an argument similar to that in the proof of Prop. 3, we can show that dR ≤ d + r . For m ≥ n and a basis Bm of GF(q m ) over GF(q), we append m − n all-zero rows to the expansion with respect to Bn of codewords in C , then the matrices are the expansions (with respect to Bm ) of codewords of a CRC of length n and rank r over GF(q m ) and minimum rank distance ≥ 2d. We now derive the bounds on the diameter DR of C . Let x and y be distinct codewords in C such that dS (C(x), C(y)) = dS (R(x), R(y)) = DS , then by Theorem 1,
1 2
max{DS , 3DS − 2r} ≤ DR . Let x′
and y′ be distinct codewords in C such that dR (x′ , y′ ) = DR , then by Theorem 1, DR ≤ 12 (DS + 2r). B. Proof of Proposition 8 Proof: For all x ∈ GF(q m−1 )m−1 with rank m − 1, define g : x 7→ y ∈ GF(q m )m such that X 0 ∈ GF(q)m×m , Y= (32) 0 1 where X and Y are the expansions of x and y, respectively. By (32), for all x, x′ ∈ GF(q m−1 )m−1 with rank m − 1, we have rk(g(x)) = rk(x) + 1 = m and rk(g(x) − g(x′ )) = rk(x − x′ ). Therefore g is a homomorphism from Kq (m − 1, m − 1, d, m − 1) to Kq (m, m, d, m). Applying (6) to these graphs, and noticing that α(m, m) = q m−1 (q m − 1)α(m − 1, m − 1), we obtain (12).
20
We now prove (13). Note that any vector x ∈ GF(q m )n with rank r belongs to
n−r 1
ELSs of
dimension n − 1. Indeed, such ELSs are those which contain E , where E is the unique ELS of dimension r such that x ∈ E . Using basic counting, there are exactly n−r such ELSs. 1
Let C be an optimal (n, d, r) CRC over GF(q m ). For all c ∈ C and all V ∈ En−1 (q m , n), we define P f (V, c) = 1 if c ∈ V and f (V, c) = 0 otherwise. For all c, V∈En−1 (qm ,n) f (V, c) = n−r 1 , and for all P V , c∈C f (V, c) = |C ∩ V|. Summing over all possible pairs, we obtain X
V∈En−1
(q m ,n)
X
X
X
f (V, c) =
V∈En−1
c∈C
X
f (V, c) =
c∈C V∈En−1 (q m ,n)
|C ∩ V|
(q m ,n)
n−r AR (q m , n, d, r). 1
[n−r 1 ] A (q m , n, d, r). The [n1 ] R restriction of C ∩ U to the ELS U [35] is an (n − 1, d, r) CRC with cardinality |C ∩ U | over GF(q m ),
Hence there exists U ∈ En−1 (q m , n) such that |C ∩ U | =
and hence
q n−r −1 m q n −1 AR (q , n, d, r)
P
c∈C
f (U, c) ≥
≤ |C ∩ U | ≤ AR (q m , n − 1, d, r).
C. Proof of Proposition 12 Proof: For all x ∈ GF(q l )k with rank s and c ∈ C , we define fr (x, c) = 1 if dR (x, c) = r P l and fr (x, c) = 0 otherwise. Note that x:rk(x)=s fr (x, c) = J(q , k, s, r, rk(c)) for all c ∈ C and P l l k c∈C fr (x, c) = |{y ∈ C − x : rk(y) = r}| ≤ AR (q , k, d, r) for all x ∈ GF(q ) . We obtain X
X
n X
fr (x, c) =
X
(33)
i=0
c∈C x:rk(x)=s
X
Ai J(q l , k, s, r, i),
fr (x, c) ≤ Ns (q l , k)AR (q l , k, d, r).
(34)
x:rk(x)=s c∈C
Combining (33) and (34), we obtain m
AR (q , n, d, r) ≥
Pn
l i=0 Ai J(q , k, s, r, i) . Ns (q l , k)
(35)
Suppose d > r + 1. For all c ∈ C , let us denote the set of vectors with rank s at distance at most def S d − r − 1 from c as Sc , and S = c∈C Sc . For x ∈ Sc , we have dR (x, c) ≤ d − r − 1 < r . We have
for c′ ∈ C and c′ 6= c, dR (x, c′ ) ≥ dR (c, c′ ) − dR (x, c) ≥ r + 1; and hence fr (x, c′ ) = 0 for all c′ ∈ C . P Therefore, c∈C fr (x, c) = 0 for all x ∈ S and h i X X XX X X fr (x, c) ≤ Ns (q l , k) − |S| AR (q l , k, d, r). (36) fr (x, c) = fr (x, c) + x:rk(x)=s c∈C
x∈S c∈C
x∈S /
rk(x)=s
c∈C
21
Since d − r − 1 < d2 , the balls with radius d − r − 1 around the codewords are disjoint and hence P P |S| = ni=0 Ai d−r−1 J(q l , k, s, t, i). Combining (33) and (36), we obtain t=0 Pn l l i=0 Ai J(q , k, s, r, i) AR (q , k, d, r) ≥ . (37) P P n l , k, s, t, i) Ns (q l , k) − i=0 Ai d−r−1 J(q t=0 Note that (35) and (37) both hold for any s and weight spectrum {Ai }. Furthermore, since AR (q l , k, d, r)
is a non-decreasing function of l and k, AR (q m , n, d, r) ≥ AR (q l , k, d, r) for all max{r, d} ≤ k ≤ n and k ≤ l ≤ m. Thus, we have (18) and (19).
D. Proof of Proposition 15 Proof: By Proposition 11, we obtain AR (q m , m, d, m) ≤ α(m, m − d + 1) for r = n = m and AR (q m , n, d, r) ≤ nr α(m, r − d + 1) < nr q m(r−d+1) otherwise. We now derive lower bounds Pr j on M (q m , n, d, r). Again, M (q m , n, d, r) = nr j=d (−1) µj where µj > µj−1 for d + 1 ≤ j ≤ r . Therefore, when needed, we shall only consider the last terms in the summation. m
−1 m First, M (q m , m, m − 1, m) = (q 2m − 1) − qq−1 (q − 1) >
q−2 2m q−1 (q
− 1) >
q−2 q−1 α(m, 2),
which leads
to (24). For q = 2, M (2m , m, m − 1, m) = 2(2m − 1) = (2m−1 − 1)−1 α(m, 2), which results in (23). Second, when r = n = m and d = m − 2, α(m, 2) α(m, 1) 2m (q − 1) + 2 (q m − 1) q−1 (q − 1)(q − 1) q−2 1 α(m, 1)(q 2m − 1) + 2 α(m, 2)(q m − 1) q−1 (q − 1)(q − 1)
M (q m , m, m − 2, m) = (q 3m − 1) − > >
(q 2 − 1)(q − 2) + 1 α(m, 1), (q 2 − 1)(q − 1)
which leads to (25). Third, when r = n = m and d < m − 2, by considering the last four terms in the summation, we obtain M (q m , m, d, m) > (q m(m−d+1) − 1) −
α(m, 1) m(m−d) (q − 1) q−1
α(m, 2) α(m, 3) (q m(m−d−1) − 1) − 3 (q m(m−d−2) − 1) (q 2 − 1)(q − 1) (q − 1)(q 2 − 1)(q − 1) q3 − 2 q−2 + α(m, m − d + 1), > q − 1 (q 3 − 1)(q 2 − 1)(q − 1)
+
which results in (26).
22
Fourth, when d < r < m, by considering the last two terms in the summation, we obtain n r m(r−d) M (q m , n, d, r) ≥ (q m(r−d+1) − 1) − (q − 1) r 1 n q m(r−d+1) − 1 − q m(r−d)+r + q r ≥ r n m(r−d+1) ≥ q (1 − q r−m ). r Therefore, since r < m, B(q m , n, d, r) < (1 − q r−m )−1 ≤
q q−1 ,
(38) (39)
which leads to (27).
E. Proof of Proposition 17 Proof: We first derive a lower bound on aR (ν, δ, ρ). We shall use the following bounds on the def Q∞ −j Gaussian polynomial: q r(n−r) ≤ nr < Kq−1 q r(n−r) , where Kq = j=1 (1 − q ) [35, Lemma 1].
For d ≤ r , Proposition 14 yields AR (q m , n, d, r) ≥ q r(n−r)+m(r−d) , which asymptotically becomes aR (ν, δ, ρ) ≥ ρ(1 + ν − ρ) − δ for δ ≤ ρ. Similarly, for d > r , Proposition 16 yields AR (q m , n, d, r) ≥ q r(n−r)+n(r−d+1) , which asymptotically becomes aR (ν, δ, ρ) ≥ ρ(2ν − ρ) − νδ for δ ≥ ρ. Also, by (5),
(9) asymptotically yields aR (ν, δ, ρ) ≥ min{(ν − ρ)(ρ − 2δ ), ρ(ν − ρ − 2δ )} for δ ≥ ρ. Proposition 5 and (5) yield logq AR (q m , n, d, r) ≥ min{(n − r)(2r − d − 2p + 1), (m − r)(p + 1)} for d > r and 2r ≤ n. Treating the two terms as functions and assuming that p is real, the lower bound is
maximized when p=
Using p =
j
(n−r)(2r−d+1)−m+r m+2n−3r
k
(n − r)(2r − d + 1) − m + r . m + 2n − 3r
, asymptotically we obtain aR (ν, δ, ρ) ≥
(40) (1−ρ)(ν−ρ) 1+2ν−3ρ (2ρ
− δ) for 2ρ ≤ ν .
For d > r and n ≤ 2r ≤ m, Proposition 5 and (5) lead to logq AR (q m , n, d, r) ≥ min{r(n − d − 2p + 1), (m − r)(p + 1)}. After maximizing this expression over p, we asymptotically obtain aR (ν, δ, ρ) ≥ ρ(1−ρ) 1+ρ (ν
− δ) for ν ≤ 2ρ ≤ 1.
For d > r and 2r ≥ m, Proposition 5 and (5) lead to logq AR (q m , n, d, r) ≥ min{r(n − d − 2p + 1), r(m − 2r + p + 1)}. After maximizing this expression over p, we asymptotically obtain aR (ν, δ, ρ) ≥ ρ 3 (2
− 4ρ + ν − δ) for 2ρ ≥ 1.
We now derive an upper bound on aR (ν, δ, ρ). First, Proposition 11 gives AR (q m , n, d, r)
arXiv:0803.2262v5 [cs.IT] 28 Jul 2008
Lehigh University, PA 18015, USA E-mails: {magc, yan}@lehigh.edu
Abstract Constant-dimension codes have recently received attention due to their significance to error control in noncoherent random network coding. What the maximal cardinality of any constant-dimension code with finite dimension and minimum distance is and how to construct the optimal constant-dimension code (or codes) that achieves the maximal cardinality both remain open research problems. In this paper, we introduce a new approach to solving these two problems. We first establish a connection between constantrank codes and constant-dimension codes. Via this connection, we show that optimal constant-dimension codes correspond to optimal constant-rank codes over sufficiently large extension fields. As such, the two aforementioned problems are equivalent to determining the maximum cardinality of constant-rank codes and to constructing optimal constant-rank codes, respectively. To this end, we derive bounds on the maximum cardinality of a constant-rank code with a given minimum rank distance, propose explicit constructions of optimal or asymptotically optimal constant-rank codes, and establish asymptotic bounds on the maximum rate of a constant-rank code.
I. I NTRODUCTION While random network coding [1]–[3] has proved to be a powerful tool for disseminating information in networks, it is highly susceptible to errors caused by various sources such as noise, malicious or malfunctioning nodes, or insufficient min-cut. If received packets are linearly combined at random to deduce the transmitted message, even a single error in one erroneous packet could render the entire transmission useless. Thus, error control for random network coding is critical and has received growing attention recently. Error control schemes proposed for random network coding assume two types of transmission models: some [4]–[8] depend on and take advantage of the underlying network topology
2
or the particular linear network coding operations performed at various network nodes; others [9], [10] assume that the transmitter and receiver have no knowledge of such channel transfer characteristics. The contrast is similar to that between coherent and noncoherent communication systems. Error control for noncoherent random network coding was first considered in [9]1 . Motivated by the property that random network coding is vector-space preserving, an operator channel that captures the essence of the noncoherent transmission model was defined in [9]. Similar to codes defined in complex Grassmannians for noncoherent multiple-antenna channels, codes defined in Grassmannians over a finite field [12], [13] and used with the subspace distance (cf. [9, (3)]) play a significant role in error control for noncoherent random network coding; Under the subspace distance, the weight of a subspace is simply its dimension; thus, we refer to these codes as constant-dimension codes (CDCs) henceforth. The standard advocated approach to random network coding (see, e.g., [2]) involves transmission of packet headers used to record the particular linear combination of the components of the message present in each received packet. From coding theoretic perspective, the set of subspaces generated by the standard approach may be viewed as a suboptimal CDC with minimum subspace distance 2 in the Grassmannian, because the Grassmannian contains more spaces with minimum subspace distance 2 than those obtained by the standard approach [9]. Hence, studying random network coding from coding theoretic perspective results in better error control schemes. General studies of subspace metric codes (also referred to as codes in projective space or projective geometry) started only recently (see, for example, [14], [15]). On the other hand, there is a steady stream of works that focuses on codes in the Grassmannian. For example, Delsarte [12] proved that the Grassmannian endowed with the subspace distance forms an association scheme, and derived its parameters. The nonexistence of perfect codes in the Grassmannian was proved in [13], [16]. In [17], it was shown that Steiner structures yield diameter-perfect codes in the Grassmannian; properties and constructions of these structures were studied in [18]; in [19], it was shown that Steiner structures result in optimal CDCs. Related work on certain intersecting families and on byte-correcting codes can be found in [20] and [21], respectively. An application of codes in the Grassmanian to linear authentication schemes was considered in [22]. In [9], a Singleton bound for CDCs and a family of codes that are nearly Singleton-bound achieving are proposed, and a recursive construction of CDCs which outperform the codes in [9] was given in [23]. Despite the asymptotic optimality of the Singleton bound and the codes proposed in [9], both are not optimal in finite cases: upper bounds tighter than the Singleton bound 1
A related work [11] considers security issues in noncoherent random network coding.
3
exist and can be achieved in some special cases [19]. It is yet to be determined the maximal cardinality of a CDC with finite dimension and minimum distance, and it is not clear how to construct the optimal code (or codes) that achieves the maximal cardinality. In this paper, we introduce a novel approach to solving the two aforementioned problems. Namely, we aim to solve these problems via rank metric codes, in particular, constant-rank codes (CRCs), which are the counterparts in rank metric codes of constant Hamming weight codes. There are several reasons for our approach. First, it is difficult to answer the two questions based on CDCs directly since the projective space lacks a natural group structure [10]. Second, the rank metric is similar to the Hamming metric and hence familiar results from the Hamming space can be readily adapted. Furthermore, there are extensive works on rank metric codes in the literature. Finally, the rank metric has been shown relevant to error control for both noncoherent [10] and coherent [24] random network coding. Based on our approach, this paper makes two main contributions. Our first main contribution is that we establish a connection between CRCs and CDCs. Via this connection, we show that optimal CDCs correspond to optimal CRCs over sufficiently large extension fields. This connection converts the aforementioned open research problems about CDCs into research problems about CRCs, thereby allowing us to use rich results in rank metric codes to tackle such problems. Since constant-rank codes have received little attention in the literature, our second main contribution is our investigation of the properties of CRCs. In particular, we derive upper and lower bounds on the maximum cardinality of a CRC, propose explicit constructions of optimal or asymptotically optimal CRCs, and establish asymptotic bounds on the maximum rate of CRCs. The rest of the paper is organized as follows. Section II reviews some necessary background. In Section III, we determine the connection between optimal CRCs and optimal CDCs. In Section IV, we study the maximum cardinality of CRCs, and we present our results on the asymptotic behavior of the maximum rate of a CRC. II. P RELIMINARIES A. Rank metric codes and elementary linear subspaces Error correction codes with the rank metric [25]–[27] have been receiving steady attention in the literature due to their applications in storage systems [27], public-key cryptosystems [28], space-time coding [29], and network coding [9], [10]. Below we review some important properties of rank metric codes established in [25]–[27].
4
Consider a vector x of length n over GF(q m ). The field GF(q m ) may be viewed as an m-dimensional vector space over GF(q). The rank weight of x, denoted as rk(x), is defined to be the maximum number of coordinates of x that are linearly independent over GF(q) [26]. For any basis Bm of GF(q m ) over GF(q), each coordinate of x can be expanded to an m-dimensional column vector over GF(q) with
respect to Bm . Thus the rank weight of x is also given by the rank of the m × n matrix over GF(q) obtained by expanding all the coordinates of x [26]. We shall assume that the expansions are with respect to a given basis Bm of GF(q m ) over GF(q) henceforth. def
For all x, y ∈ GF(q m )n , it is easily verified that dR (x, y) = rk(x − y) is a metric over GF(q m )n , referred to as the rank metric henceforth [26]. The minimum rank distance and rank diameter of a code C , denoted as dR and DR , respectively, are simply the minimum and maximum rank distance over all
possible pairs of distinct codewords. It is shown in [25]–[27] that the minimum rank distance of a block code of length n and cardinality M over GF(q m ) satisfies dR ≤ n − logqm M + 1. In this paper, we refer to this bound as the Singleton
bound for rank metric codes and codes that attain the equality as maximum rank distance (MRD) codes. We refer to the subclass of linear MRD codes introduced in [30] as generalized Gabidulin codes. We denote the number of vectors of rank r (0 ≤ r ≤ min{m, n}) in GF(q m )n as Nr (q m , n) = n def n def def Qr−1 m i i=0 (q − q ), and r = α(n, r)/α(r, r) for r ≥ 1. r α(m, r) [26], where α(m, 0) = 1, α(m, r) = The nr term is often referred to as a Gaussian polynomial [31]. The volume of a ball with rank radius P r in GF(q m )n is denoted as Vr (q m , n) = ri=0 Ni (q m , n). We denote the intersection of two spheres in GF(q m )n of radii r and s, with distance between their centers d as J(q m , n, r, s, d). We will omit the
dependence of the quantities defined above on q m and n when there is no ambiguity about the vector space considered. By [32, Theorem 3.6], we obtain n X mn Nl Pr (l)Ps (l)Pd (l), q Nd J(r, s, d) =
(1)
l=0
where Pj (l) is a q -Krawtchouk polynomial [33], [34]: n X n−i j−i im+(j−i)(j−i−1)/2 n − l (−1) q Pj (l) = . n−j l
(2)
i=0
For all q , 1 ≤ d ≤ r ≤ n ≤ m, the number of codewords of rank r in an (n, n − d + 1, d) linear MRD code over GF(q m ) is given by [26] X r def n r−j r (r−j)(r−j−1)/2 m (−1) M (q , n, d, r) = q m(j−d+1) − 1 . q r j
(3)
j=d
An elementary linear subspace (ELS) [35] is defined to be a linear subspace V ⊆ GF(q m )n for which there exists a basis of vectors in GF(q)n . We denote the set of all ELSs of GF(q m )n with dimension
5
v as Ev (q m , n) and the set of all ELSs as E(q m , n). It can be easily shown that |Ev (q m , n)| =
n v
for
all m. An ELS has properties similar to those for a set of coordinates [35]. In particular, any vector belonging to an ELS with dimension r has rank no more than r ; conversely, any vector x ∈ GF(q m )n with rank r belongs to a unique ELS in Er (q m , n). B. Subspace distance and constant-dimension codes For two subspaces of GF(q)n , U and V , it is easily verified that def
dS (U, V) = dim(U + V) − dim(U ∩ V) = 2 dim(U + V) − dim(U) − dim(V)
(4)
is a metric over E(q, n), referred to as the subspace metric [9] henceforth. The subspace distance between U and V thus satisfies dS (U, V) = 2rk(XT Y T )−rk(X)−rk(Y), where X and Y are generator matrices
of U and V , respectively. The minimum subspace distance and subspace diameter of any subspace metric code, denoted as dS and DS , respectively, are the minimum and maximum subspace distance over all possible pairs of distinct subspaces. A constant-dimension code Ω of length n and constant-dimension r over GF(q) is defined to be a nonempty subset of Er (q, n) [9]. When Ω has minimum subspace distance dS , we refer to Ω as an (n, dS , r) CDC over GF(q) and we denote the maximum cardinality of all (n, dS , r) CDCs over GF(q)
as AS (q, n, dS , r). Since AS (q, n, dS , r) = AS (q, n, dS , n − r) [19], only the case where 2r ≤ n needs to be considered. By (4), it is easily verified that the subspace distance between any two subspaces of the same dimension is even; thus, the minimum distance of any CDC is even. Hence, we shall consider AS (q, n, 2d, r) for 2 ≤ d ≤ r henceforth since AS (q, n, 2, r) = nr and AS (q, n, 2d, r) = 1 for d > r .
Upper and lower bounds on AS (q, n, 2d, r) were derived in [9], [14], [15], [19], [22]. In particular, for all q , 2r ≤ n, and 2 ≤ d ≤ r , it was shown in [9], [22] that q (n−r)(r−d+1) ≤ AS (q, n, 2d, r) ≤
α(n, r − d + 1) . α(r, r − d + 1)
(5)
Note that the lower bound is both constructive and asymptotically tight [9], and we use this bound in our derivation of asymptotic rate of CRCs. C. Preliminary graph-theoretic results We review some results in graph theory given in [36]. We denote the set of vertices in a graph G as V (G), and the adjacency between any two vertices u and v as u ∼ v .
Definition 1: Let G and H be two graphs. A mapping f from V (G) to V (H) is a homomorphism if for all u, v ∈ V (G), f (u) ∼ f (v) if u ∼ v .
6
Definition 2: Let G be a graph and φ a bijection from V (G) to itself. φ is called an automorphism of G if for all u, v ∈ V (G), φ(u) ∼ φ(v) if and only if u ∼ v . Definition 3: The graph G is vertex transitive if for all u, v ∈ V (G), there exists an automorphism φ of G such that φ(u) = v . An independent set of a graph G is a subset of V (G) with no adjacent vertices. The independence number α(G) of G is the maximum cardinality of an independent set of G. If H is a vertex transitive graph and if there is a homomorphism from G to H , then [36], [37] α(G) ≥ α(H)
III. C ONNECTION
|G| . |H|
(6)
BETWEEN CONSTANT- DIMENSION CODES AND CONSTANT- RANK CODES
In this section, we first establish some connections between the rank metric and the subspace metric. We then define constant-rank codes and we show how optimal constant-rank codes can be used to construct optimal CDCs. For x ∈ GF(q m )n and a basis Bm of GF(q m ) over GF(q), let us consider X ∈ GF(q)m×n , the expansion of x with respect to Bm . Let us denote the row span and the column span of X over GF(q) as R(x; Bm ) and C(x; Bm ), respectively. Clearly, C(x; Bm ) ∈ Er (q, m) and R(x; Bm ) ∈ Er (q, n), where r = rk(x). We remark that R(x; Bm ) does not depend on Bm since changing a basis simply results in
elementary row operations on X; on the other hand, C(x; Bm ) depends on Bm , even the order of the elements in Bm , although the dimension of C(x; Bm ) is independent of Bm . We shall henceforth assume that the basis and the order of the basis elements are fixed and simply use the notations R(x) and C(x). def
The notations introduced above are naturally extended to codes as follows: for C ⊆ GF(q m )n , C(C) = def
{C(c) : c ∈ C} and R(C) = {R(c) : c ∈ C}.
Lemma 1: For U ∈ Er (q, m), V ∈ Er (q, n), and x ∈ GF(q m )n with rank r , C(x) = U and R(x) = V if and only if X = GT H, where G ∈ GF(q)r×m is a generator matrix of U and H ∈ GF(q)r×n is a generator matrix of V . The proof of Lemma 1 is straightforward and hence omitted. We remark that X = GT H is referred to as a rank factorization [38]. Alternatively, x = bH, where the expansion (with respect to a basis of GF(q m ) over GF(q)) of b ∈ GF(q m )r is given by GT . We now derive a relation between the rank
distance between two vectors and the subspace distances between their respective row and column spans. Theorem 1: For all x, y ∈ GF(q m )n , let us denote dS (C(x), C(y)) and dS (R(x), R(y)) as dC and dR , respectively. Without loss of generality, assume dC ≥ dR , then 1 1 max{dC + |rk(x) − rk(y)|, 2dC + dR − rk(x) − rk(y)} ≤ dR (x, y) ≤ [dR + rk(x) + rk(y)] . (7) 2 2
7
Proof: We have R(x − y) ⊆ R(x) + R(y) and hence dR (x, y) = dim R(x − y) ≤ dim(R(x) + R(y)) = 12 (dR + rk(x) + rk(y)) by (4).
Let x = bH and y = b′ H′ so that b ∈ GF(q m )rk(x) and b′ ∈ GF(q m )rk(y) . By definition of the subspace distance, dim(C(x) ∩ C(y)) = 1 2
1 2
[rk(x) + rk(y) − dC ]. Therefore, we can select d1 =
[rk(x) − rk(y) + dC ] coordinates β0 , β1 , . . . , βd1 −1 of b linearly independent to b′ , and d2 = 12 [rk(y)−
rk(x) + dC ] coordinates β0′ , β1′ , . . . , βd′ 2 −1 of b′ linearly independent to b. By (4), it can be easily shown
that dC ≥ |rk(x) − rk(y)|. Let us select {γdC , γdC +1 , . . . , γm−1 } in GF(q m ) so that γ = {β0 , β1 , . . . , βd1 −1 , β0′ , β1′ , . . . , βd′ 2 −1 , γdC , γdC +1 , . . . , γm−1 }
constitutes a basis of GF(q m ) over GF(q). ¯T H ¯ ′T , where H ¯ and H ¯ ′ are the d1 Expanding x − y with respect to γ , we obtain rk(x − y) ≥ rk H
rows of H and the d2 rows of H′ corresponding to β0 , β1 , . . . , βd1 −1 and β0′ , β1′ , . . . , βd′ 2 −1 , respectively. ¯ rk(H ¯ ′ )} = max{d1 , d2 }. Second, since (HT H′T ) has rank ¯T H ¯ ′T ≥ max{rk(H), First, we have rk H 1 ¯ T ¯ ′T ≥ dC + 1 (dR − rk(x) − rk(y)). 2 (dR + rk(x) + rk(y)) by (4), rk H H 2
Definition 4: A constant-rank code of length n and constant-rank r over GF(q m ) is a nonempty subset
of GF(q m )n such that all elements have rank weight r . We denote a CRC with length n, minimum rank distance d, and constant-rank r as an (n, d, r) CRC over GF(q m ). Proposition 1 below shows how a CRC leads to two CDCs with their minimum subspace distance and subspace diameter related to the minimum rank distance and rank diameter of the CRC. Proposition 1: Let C be an (n, dR , r) CRC over GF(q m ) with rank diameter DR . Then R(C) ⊆ Er (q, n) is a CDC with minimum subspace distance dR ≥ 2(dR − r). Similarly, C(C) ⊆ Er (q, m) is a
CDC with minimum subspace distance dC ≥ 2(dR − r). If we denote the subspace diameters of R(C) and C(C) as DR and DC , respectively, then max{DR , DC } ≤ min{2DR , DR + r}. Proof: By definition, R(C) ⊆ Er (q, n) and C(C) ⊆ Er (q, m). Let x and y be two distinct codewords in C so that dS (R(x), R(y)) = dR . By Theorem 1, dR = dS (R(x), R(y)) ≥ 2(dR (x, y)−r) ≥ 2(dR − r), and similarly dC ≥ 2(dR − r).
Let x0 and y0 be two codewords in C such that dS (C(x0 ), C(y0 )) = DC . By Theorem 1, we have DC ≤ 2dR (x0 , y0 ) ≤ 2DR and 2DC ≤ 2dR (x0 , y0 ) + 2r − dS (R(x0 ), R(y0 )) ≤ 2DR + 2r . Let x1 and y1 be two codewords in C such that dS (R(x1 ), R(y1 )) = DR , then by Theorem 1, DR ≤ 2DR and 2DR ≤ dR (x0 , y0 ) + 2r − dS (C(x1 ), C(y1 )) ≤ 2DR + 2r .
When the minimum rank distance of a CRC is no less than its rank weight, Proposition 2 below shows how the CRC leads to two CDCs with the same cardinality, and the relations between their distances
8
can be further strengthened. Proposition 2: If C is an (n, d + r, r) CRC over GF(q m ) (2 ≤ d ≤ r ) with rank diameter DR , then R(C) ⊆ Er (q, n) is a CDC with cardinality |C| and minimum subspace distance dR ≥ 2d. Similarly, C(C) ⊆ Er (q, m) is a CDC with cardinality |C| and minimum subspace distance dC ≥ 2d. We also have 1 2
max{2dC + dR − 2r, 2dR + dC − 2r} ≤ d + r . If we denote the subspace diameters of R(C) and C(C)
as DR and DC , respectively, then max{DR , DC } ≤ 2DR , min{DR , DC } ≥ 2DR − 2r , and 1 max{DC + 2dR , DR + 2dC , 2DC + dR , 2DR + dC } ≤ DR + r. 2 Proof: Let x and y be any two distinct codewords in C . By Theorem 1, dS (R(x), R(y)) ≥ 2dR (x, y) − 2r ≥ 2d > 0, and hence dR ≥ 2d and |R(C)| = |C|. Similarly, dS (C(x), C(y)) ≥ 2d > 0,
and thus dC ≥ 2d and |C(C)| = |C|. Furthermore, if dR (x, y) = d + r , then by Theorem 1, 2(d + r) ≥ 2dS (C(x), C(y)) + dS (R(x), R(y)) − 2r ≥ 2dC + dR − 2r . Similarly, we obtain 2(d + r) ≥ 2dR + dC − 2r .
We now prove the inequalities involving DC . Let x0 and y0 be two codewords in C such that DC = dS (C(x0 ), C(y0 )). By Theorem 1, we have DC ≤ 2dR (x0 , y0 ) ≤ 2DR , DC ≤ 2dR (x0 , y0 ) + 2r − 2dS (R(x0 ), R(y0 )) ≤ 2DR + 2r − 2dR , and 2DC ≤ 2dR (x0 , y0 ) + 2r − dS (R(x0 ), R(y0 )) ≤ 2DR + 2r − dR . Let x1 and y1 be two codewords in C such that dR (x1 , y1 ) = DR , then by Theorem 1, DR ≤
1 2
[dS (C(x1 ), C(y1 )) + 2r] ≤
1 2
[DC + 2r]. The other inequalities involving DR are obtained
similarly. Using Lemma 1, we can construct CRCs in GF(q m )n from a pair of CDCs in GF(q)n and GF(q)m , respectively. Proposition 3: Let Γ be an (m, dC , r) CDC over GF(q) and ∆ be an (n, dR , r) CDC over GF(q). Then there exists a CRC C with length n, constant-rank r , and cardinality min{|Γ|, |∆|} over GF(q m ) satisfying C(C) ⊆ Γ and R(C) ⊆ ∆. Furthermore, its minimum distance dR satisfies dR − 2r, 0} ≤ dR ≤
1 2
1 2
max{dC , dR } + 12 max{dC +
max{dC , dR } + r . If we denote the subspace diameters of ∆ and Γ as DR and
DC , respectively, then the rank diameter DR of C satisfies
1 2
min{DR , DC } ≤ DR ≤
1 2
min{DR , DC } + r
and min{DC + 2dR , DR + 2dC , 2DC + dR , 2DR + dC } ≤ 2DR + 2r. Proof: Denote the generator matrices of the component subspaces of Γ and ∆ as Gi and Hi , def
respectively. Define the code C formed by the codewords ci = bi Hi for 0 ≤ i ≤ min{|Γ|, |∆|} − 1, where the expansion of bi ∈ GF(q m )r is given by GTi . Then C(C) ⊆ Γ and R(C) ⊆ ∆ by Lemma 1 and the lower bound on dR follows Theorem 1. By construction, C(C) = Γ or R(C) = ∆. If C(C) = Γ, then let x and y be distinct codewords in C such that dS (C(x), C(y)) = dC . By Theorem 1, we obtain dR ≤ dR (x, y) ≤ 12 (dC + 2r). Similarly, if
9
R(C) = ∆, we obtain dR ≤ 21 (dR + 2r). Combining these results, we obtain dR ≤
1 2
max{dR + 2r, dC +
2r}.
Let x and y be two codewords in C such that dR (x, y) = DR , then by Theorem 1, 2DR ≤ dS (R(x), R(y))+ 2r ≤ DS (R(C)) + 2r ≤ DR + 2r , and similarly 2DR ≤ DC + 2r . If C(C) = Γ, then let x′ and y′ be
distinct codewords in C such that dS (C(x), C(y)) = DC ; by Theorem 1, R(C) = ∆, we obtain 21 DR ≤ DR , and hence
1 2
1 2 DC
≤ dR (x′ , y′ ) ≤ DR . If
min {DC , DR } ≤ DR . By a similar argument, we obtain
min{DC + 2dR , DR + 2dC , 2DC + dR , 2DR + dC } ≤ 2DR + 2r.
Proposition 4: Let ∆ be an (n, 2d, r) CDC over GF(q) with subspace diameter DS . Then for any extension field GF(q m ) with m ≥ n, there exists a CRC C of length n and constant-rank r , over GF(q m ) such that R(C) = ∆ and |C| = |∆|. The minimum distance dR of C satisfies 2d ≤ dR ≤ d + r
and the rank diameter DR of C satisfies
1 2
max{DS , 3DS − 2r} ≤ DR ≤ 12 DS + r .
The proof of Proposition 4 follows Theorem 1, and is similar to that of Proposition 3. The proof is given in Appendix A. We denote the maximum cardinality of an (n, d, r) CRC over GF(q m ) as AR (q m , n, d, r). If C is an (n, d, r) CRC over GF(q m ), then the code obtained by transposing all the expansion matrices of codewords in C forms an (m, d, r) CRC over GF(q n ) with the same cardinality, and vice versa2 . Therefore AR (q m , n, d, r) = AR (q n , m, d, r), and henceforth in this paper we assume n ≤ m without loss of
generality. We further observe that AR (q m , n, d, r) is a non-decreasing function of m and n, and a nonincreasing function of d, and that AS (q, n, 2d, r) is a non-decreasing function of n and a non-increasing function of d. Proposition 5: For all q , 2 ≤ d ≤ r ≤ n ≤ m, and any 0 ≤ p ≤ r , min{AS (q, n, 2(d + 2p), r), AS (q, m, 2(r − p), r)} ≤ AR (q m , n, d + r, r) ≤ AS (q, n, 2d, r).
(8)
AR (q m , n, d + r, r) ≥ AS (q, n, d + r, r).
(9)
Also,
Proof: Using the monotone properties of AR (q m , n, dR , r) and AS (q, n, dS , r) above, the upper bound in (8) follows Proposition 2, while the lower bound in (8) follows Proposition 3 for dC = 2(r − p) and dR = 2(d + 2p). Finally, (9) follows Proposition 4 and the monotone properties of AR (q m , n, dR , r).
We remark that the lower bound in (8) is trivial for d + 2p > min{r, n − r} or r − p > min{r, m − r}. Therefore, the lower bound in (8) is nontrivial when max{0, 2r − m} ≤ p ≤ 2
However, the linearity of codes is not preserved through transposition.
1 2
min{r − d, n − r − d}.
10
Combining the bounds in (8), we obtain that the cardinalities of optimal CRCs over sufficiently large fields are equal to the cardinalities of CDCs with related distances. Furthermore, we show that optimal CDCs can be constructed from such optimal CRCs. Theorem 2: For all q , 2r ≤ n ≤ m, and 1 ≤ d ≤ r , AR (q m , n, d + r, r) = AS (q, n, 2d, r) if either d = r or m ≥ m0 , where m0 = (n − r)(r − d + 1) + r + 1. Furthermore, if C is an (n, d + r, r) optimal
CRC over GF(q m ) for m ≥ m0 or d = r , then R(C) is an optimal (n, 2d, r) CDC over GF(q). Proof: First, the case where d = r directly follows (8) for p = 0. Second, if d < r and m ≥ m0 , by (5) we obtain AS (q, m, 2r, r) ≥ q m−r ≥ q m0 −r . Also, by [35, Lemma 1], we obtain q r(r−d+1)−1 < α(r, r − d + 1) ≤ q r(r−d+1) for all 2 ≤ d < r , and hence (5) yields AS (q, n, 2d, r) < q (n−r)(r−d+1)+1 = q m0 −r ≤ AS (q, m, 2r, r). Thus, when p = 0, the lower bound in (8) simplifies to AR (q m , n, d + r, r) ≥ AS (q, n, 2d, r). Combining with the upper bound in (8), we obtain AR (q m , n, d + r, r) = AS (q, n, 2d, r).
The second claim immediately follows Proposition 2. Theorem 2 implies that to determine AS (q, n, 2d, r) and to construct optimal CDCs, it is sufficient to determine AR (q m , n, d + r, r) and to construct optimal CRCs over an extension field sufficiently large. We observe that this implies that AR (q m , n, d + r, r) remains constant for all m ≥ m0 . When d = r , AR (q m , n, 2r, r) remains constant for m ≥ n. When d = 1, m0 = (n − r + 1)r + 1, but AR (q m , n, r + 1, r)
remains constant for m ≥ n, and this is shown in Section IV-C. IV. C ONSTANT- RANK
CODES
Having proved that optimal CRCs over sufficiently large extension fields lead to optimal CDCs, in this section we investigate the properties of CRCs. A. Graph-theoretic results for constant-rank codes We now define two families of graphs which are instrumental in our analysis of CRCs. Definition 5: The bilinear forms graph Rq (m, n, d) has as vertices all the vectors in GF(q m )n and two vertices x and y are adjacent if and only if dR (x, y) < d. The constant-rank graph Kq (m, n, d, r) is the subgraph of Rq (m, n, d) induced by the vectors in GF(q m )n with rank r . The orders of the bilinear forms and constant-rank graphs are thus given by |Rq (m, n, d)| = q mn and |Kq (m, n, d, r)| = Nr (q m , n). An independent set of Rq (m, n, d) corresponds to a code with minimum rank distance ≥ d. Due to the existence of MRD codes for all parameter values [39], we have α(Rq (m, n, d)) = q m(n−d+1) . Similarly, an independent set of Kq (m, n, d, r) corresponds to a CRC with minimum rank distance ≥ d, and hence α(Kq (m, n, d, r)) = AR (q m , n, d, r).
11
Lemma 2: The bilinear forms graph Rq (m, n, d) is vertex transitive for all q , m, n, and d. The constantrank graph Kq (m, m, d, m) is vertex transitive for all q , m, and d. Proof: For given u, v ∈ GF(q m )n , define φ(x) = x + v − u for all x ∈ GF(q m )n . It is easily shown that φ is a graph automorphism of Rq (m, n, d) satisfying φ(u) = v. By Definition 3, Rq (m, n, d) is hence vertex transitive. Let u, v ∈ GF(q m )m have rank m, and denote their expansions as U and V, respectively. For all x ∈ GF(q m )m with rank m, define φ(x) = y such that Y = XU−1 V, where X and Y are the
expansions of x and y, respectively. We have φ(u) = v, rk(φ(x)) = m, and for all x, z ∈ GF(q m )m , dR (φ(x), φ(z)) = rk(XU−1 V−ZU−1 V) = rk(X−Z) = dR (x, z). By Definition 2, φ is an automorphism
which maps u to v and hence Kq (m, m, d, m) is vertex transitive. It is worth noting that Kq (m, n, d, r) is not vertex transitive in general. B. Bounds We now derive bounds on the maximum cardinality of CRCs. We first remark that the bounds on AR (q m , n, d, r) derived in Section III can be used in this section. Also, since AR (q m , n, 1, r) = Nr (q m , n)
and AR (q m , n, d, r) = 1 for d > 2r , we shall assume 2 ≤ d ≤ 2r henceforth. We first derive the counterparts of the Gilbert and the Hamming bounds for CRCs in terms of intersections of spheres with rank radii. def
Proposition 6: For all q , 1 ≤ r, d ≤ n ≤ m, and t = ⌊ d−1 2 ⌋, Nr (q m , n) Nr (q m , n) ≤ AR (q m , n, d, r) ≤ Pt . (10) Pd−1 m m i=0 J(q , n, i, r, r) i=0 J(q , n, i, r, r) The proof is straightforward and hence omitted. The Hamming bound is generalized as follows. def
Proposition 7: For all q , 1 ≤ r, d, s ≤ n ≤ m, and t = ⌊ d−1 2 ⌋, Ns (q m , n) . (11) m i=0 J(q , n, i, s, r) be an (n, d, r) CRC over GF(q m ). For all 0 ≤ k ≤ K − 1 and
AR (q m , n, d, r) ≤ Pt
K−1 Proof: Let C = {ck }k=0
0 ≤ s ≤ n − 1, if we denote the set of vectors in GF(q m )n with rank s and distance ≤ t from ck Pt as Rk,s , then |Rk,s | = i=0 J(i, s, r) for all k . Clearly Rk,s ∩ Rl,s = ∅ for all k 6= l, and hence SK−1 Ns ≥ | k=0 Rk,s | = K|Rk,s |, which yields (11).
We now derive upper bounds on AR (q m , n, d, r). We begin by proving the counterpart in rank metric
codes of a well-known bound on constant-weight codes proved by Johnson in [40].
12
Proposition 8: For all q , 1 ≤ r, d < n ≤ m, AR (q m , m, d, m) ≤ q m−1 (q m − 1)AR (q m−1 , m − 1, d, m − 1) qn − 1 AR (q m , n − 1, d, r). q n−r − 1 The proof of Proposition 8 is given in Appendix B. AR (q m , n, d, r) ≤
(12) (13)
The Singleton bound for rank metric codes yields upper bounds on AR (q m , n, d, r). For any I ⊆ {0, 1, . . . , n}, let AR (q m , n, d, I) denote the maximum cardinality of a code of length n and minimum
rank distance d over GF(q m ) such that all codewords have rank weights belonging to I . def
Proposition 9: For all q , 1 ≤ r, d ≤ n ≤ m, define Pr = {i : 0 ≤ i ≤ n, |i − r| ≥ d} and def
Qr,a = Pr ∩ {a + kd : k ∈ Z} for all 0 ≤ a < d. Then AR (q m , n, d, r) ≤ q m(n−d+1) − AR (q m , n, d, Pr ) X X AR (q m , n, d, j) . M (q m , n, d, i), ≤ q m(n−d+1) − max
(14) (15)
j∈Qr,a
i∈Pr
Proof: For 0 ≤ j ≤ n, let Cj be an optimal (n, d, j) CRC and let C ⊆ GF(q m )n be a code with minimum rank distance d and whose codewords have rank weights belonging to Pr . Eq. (14) directly ˙ r , where ∪˙ denotes disjoint union. Let G be an (n, n − d + 1, d) follows the Singleton bound on C ∪C
linear MRD code over GF(q m ), and denote the subset of codewords with ranks belonging to Pr as G′ . def S Finally define C ′ = ˙ j∈Qr,a Cj . Both G′ and C ′ are codes in GF(q m )n with minimum rank distance d and whose codewords have rank weights belonging to Pr , hence AR (q m , n, Pr ) ≥ max{|G′ |, |C ′ |},
which leads to (15). We now determine the counterpart of the Singleton bound for CRCs. def
Proposition 10: For all 0 ≤ i ≤ min{d − 1, r}, Ji = {r − i, r − i + 1, . . . , min{n − i, r}}. Then AR (q m , n, d, r) ≤ AR (q m , n − i, d − i, Ji )
(16)
min{n−i,r}
≤
X
AR (q m , n − i, d − i, j).
(17)
j=r−i
Proof: Let C be an optimal (n, d, r) CRC over GF(q m ), and consider the code Ci obtained by puncturing i coordinates of the codewords in C . Since i ≤ r , the codewords of Ci all have ranks between r − i and min{n − i, r}. Also, since i < d, any two codewords have distinct puncturings, and we obtain |Ci | = |C| and dR (Ci ) ≥ d − i. Hence AR (q m , n, d, r) = |C| = |Ci | ≤ AR (q m , n − i, d − i, Ji ). (17)
directly follows a union bound on AR (q m , n − i, d − i, Ji ). We now combine the counterparts of the Johnson bound in (13) and of the Singleton bound in Proposition 10 in order to obtain an upper bound on AR (q m , n, d, r) for d ≤ r .
13
Proposition 11: For all q , 1 ≤ d ≤ r ≤ n ≤ m, AR (q m , n, d, r) ≤ Proof: Applying (13) n − r times successively, we obtain AR
n r
α(m, r − d + 1). ≤ nr AR (q m , r, d, r). For
(q m , n, d, r)
n = r and i = d − 1, Ji = {r − d + 1} and hence (16) yields AR (q m , r, d, r) ≤ AR (q m , r − d + 1, 1, r − d + 1) = Nr−d+1 (q m , r − d + 1) = α(m, r − d + 1). Thus AR (q m , n, d, r) ≤ nr α(m, r − d + 1).
We now derive the counterpart in rank metric codes of the Bassalygo-Elias bound [41]. We also tighten
the bound when d > r + 1. Proposition 12: For max{r, d} ≤ k ≤ n, 0 ≤ s ≤ k, k ≤ l ≤ m, and any code C ⊆ GF(q l )k with def
minimum rank distance d and rank weight distribution Ai = |{c ∈ C : rk(c) = i}|, Pn l m i=0 Ai J(q , k, s, r, i) . AR (q , n, d, r) ≥ max Ns (q l , k) s,{Ai },k,l
(18)
Furthermore, if r + 1 < d ≤ 2r , then m
AR (q , n, d, r) ≥
max
Pn
Ns (q l , k) − The proof of Proposition 12 is given in Appendix C. s,{Ai },k,l
Ai J(q l , k, s, r, i) . Pd−r−1 J(q l , k, s, t, i) i=0 Ai t=0
i=0 P n
(19)
Although the RHS of (18) and (19) can be maximized over {Ai }, it is difficult to do so since {Ai } is not available for most rank metric codes with the exception of linear MRD codes. Thus, we derive a bound using the rank weight distribution of linear MRD codes. Corollary 1: For all q , 1 ≤ r, d ≤ n ≤ m, AR (q m , n, d, r) ≥ Nr (q m , n)q m(−d+1) .
(20)
Proof: Applying (18) to an (n, n−d+1, d) MRD code over GF(q m ), we obtain Ns (q m , n)AR (q m , n, d, r) ≥ Pn Pn m m i=0 M (q , n, d, i)J(q , n, s, r, i). Summing for all 0 ≤ s ≤ n, we obtain (20) since s=0 J(s, r, i) =
Nr (q m , n).
We also give an alternate proof of (20) based on the results in Section IV-A. Since Kq (m, n, d, r) is a subgraph of Rq (m, n, d), the inclusion map is a trivial homomorphism from Kq (m, n, d, r) to Rq (m, n, d). By Lemma 2, Rq (m, n, d) is vertex transitive. We hence apply (6) to these graphs, which yields (20). The RHS of (18) and (19) decrease rapidly with increasing d, rendering the bounds trivial for d approaching 2r . We investigate below the tightness of the bound in Corollary 1. def
Proposition 13: For all q , 2 ≤ d ≤ r ≤ n ≤ m, let C(q m , n, d, r) = AR (q m , n, d, r)/[Nr (q m , n)q m(−d+1) ]. Then C(q m , n, d, r) ≤ C(q m , n, d, r)
C(q m , n, d, r) ≤
q−1 −1 q Kq .
Finally,
all 1 ≤ l ≤ n − 1 [35, Lemma 1], we obtain for 2l ≤ n [35, Lemma 1] yields (21).
It is worth noting that Kq above represents the fraction of invertible m × m matrices over GF(q) as m approaches infinity. Kq−1 decreases with q and satisfies 1 < Kq−1 ≤ K2−1 < 4. Thus the bound in Corollary 1 is tight up to a scalar when d ≤ r . We also remark that the bound in (21) is tighter than that in (22). However, these bounds are not constructive. Below we derive constructive bounds on AR (q m , n, d, r).
C. Constructive bounds We now give explicit constructions of good CRCs when d ≤ r , which in turn yield asymptotically tight lower bounds on AR (q m , n, d, r). Proposition 14: For all q , 2 ≤ d ≤ r ≤ n ≤ m, AR (q m , n, d, r) ≥ M (q m , n, d, r) >
n r
q m(r−d) .
Proof: The codewords of rank r in an (n, n − d + 1, d) linear MRD code over GF(q m ) form an (n, d, r) CRC. Thus, AR (q m , n, d, r) ≥ M (q m , n, d, r). We now prove the lower bound on M (q m , n, d, r). First, for d = r , M (q m , n, r, r) = nr (q m − 1) > nr . Second, suppose d < r . By (3), M (q m , n, d, r) can be expressed as M (q m , n, d, r) = n Pr r−j µ , where µ def (r−j)(r−j−1)/2 r (q m(j−d+1) − 1). It can be easily shown that µ > (−1) = q j j j j=d r j n µj−1 for d + 1 ≤ j ≤ r , and hence M (q m , n, d, r) ≥ r (µr − µr−1 ). Therefore, M (q m , n, d, r) ≥ n m(r−d) r m(r−d) n m(r−d+1) . (q − 1)] > [(q − 1) − r q 1 r Corollary 2: For all q , 1 ≤ r ≤ n ≤ m, AR (q m , n, r, r) = nr (q m − 1). Proof: By Proposition 11, AR (q m , n, r, r) ≤ nr (q m − 1), and by Proposition 14, AR (q m , n, r, r) ≥ M (q m , n, r, r) = nr (q m − 1).
By Corollary 2, the codewords of rank r in an (n, n − r + 1, r) linear MRD code are optimal. We
investigate below the tightness of the constructive lower bound in Proposition 14. def
Proposition 15: For all q , 1 ≤ d < r ≤ n ≤ m, let B(q m , n, d, r) = AR (q m , n, d, r)/M (q m , n, d, r).
15
Then for m ≥ 3, B(2m , m, m − 1, m) ≤ 2m−1 − 1 B(q m , m, m − 1, m) < B(q m , m, m − 2, m)
2 q−2 (q 2 − 1)(q − 1) (q 2 − 1)(q − 2) + 1
(q 3 − 1)(q 2 − 1)(q − 1) (q 3 − 1)(q 2 − 1)(q − 2) + q 3 − 2 q B(q m , n, d, r) < for r < m. q−1 The proof of Proposition 15 is given in Appendix D. B(q m , m, d, m)
r using generalized Gabidulin codes [30]. Let g ∈ GF(q m )n have rank n, and for 0 ≤ i ≤ m − 1, denote the vector in GF(q m )n obtained by elevating each coordinate of g to the q ai -th power as g[i] , where a and m are coprime. Let C be the (n, n − d + 1, d) generalized T T T T , and C ′ be the (n, d − r, n − d + Gabidulin code over GF(q m ) generated by g[0] g[1] · · · g[n−d] T T T T . We consider the r + 1) generalized Gabidulin code generated by g[n−d+1] g[n−d+2] · · · g[n−r] coset C + c′ , where c′ ∈ C ′ , and we denote the number of codewords of rank r in C + c′ as σr (c′ ). Lemma 3: For all d > r , there exists c′ ∈ C ′ such that σr (c′ ) ≥ nr q m(r−d+1) .
Proof: Any codeword c′ ∈ C ′ can be expressed as c′ = cn−d+1 g[n−d+1] + cn−d+2 g[n−d+2] + . . . +
cn−r g[n−r] , where ci ∈ GF(q m ) for n − d + 1 ≤ i ≤ n − r . If cn−r = 0, then (C + c′ ) ⊂ D , where D is T T T T . Therefore the (n, n − r, r + 1) generalized Gabidulin code generated by g[0] g[1] · · · g[n−r−1] σr (c′ ) = 0 if cn−r = 0.
S Denote the number of codewords of rank r in C ⊕ C ′ as τr . Since c′ ∈C ′ (C + c′ ) = C ⊕ C ′ , we have P τr = c′ ∈C ′ σr (c′ ). Also, C ⊕ C ′ forms an (n, n − r + 1, r) MRD code, and hence τr = M (q m , n, r, r) = n m(r−d+1) n m P ′ ′ ′ ′ . Then τr = c′ :cn−r 6=0 σr (c ) < r (q − 1). Suppose that for all c ∈ C , σr (c ) < r q n m n m r (q − 1), which contradicts τr = r (q − 1).
Although Lemma 3 proves the existence of a vector c′ for which the translate C+c′ has high cardinality,
it does not indicate how to choose c′ . For d = r + 1, it can be shown that all c′ ∈ C ′ satisfy the bound, and that they all lead to optimal codes. Corollary 3: If d = r + 1, then σr (c′ ) =
n r
for all c′ ∈ C ′ .
Proof: First, by Proposition 5, σr (c′ ) ≤ AR (q m , n, r + 1, r) ≤ AS (q, n, 2, r) = nr for all c′ ∈ C ′ . Suppose there exists c′ such that σr (c′ ) < nr . Then τr < nr (q m −1), which contradicts τr = nr (q m −1).
16
Proposition 16: For all q , 1 ≤ r < d ≤ n ≤ m, AR (q m , n, d, r) ≥ that satisfy this bound can be constructed from Lemma 3.
n r
q n(r−d+1) , and a class of codes
Proof: The codewords of rank r in a code considered in Lemma 3 form an (n, d, r) CRC over GF(q m ) with cardinality ≥ nr q m(r−d+1) . Therefore, AR (q m , n, d, r) ≥ nr q m(r−d+1) . The proof is concluded by noting that AR (q m , n, d, r) ≥ AR (q n , n, d, r) ≥ nr q n(r−d+1) . Corollary 4: For all q , 1 ≤ r < n ≤ m, AR (q m , n, r + 1, r) = nr = AS (q, n, 2, r). Proof: Combine Proposition 5 and Proposition 16. We note that nr is independent of m. We also remark that the lower bound in Proposition 16 is also
trivial for d approaching 2r . Since the proof is only partly constructive, computer search can be used to help find better results for small parameter values. D. Asymptotic results
In this section, we study the asymptotic behavior of CRCs using the following set of normalized n m,
r m,
d m.
By definition, 0 ≤ ρ, δ ≤ ν , and since we assume n ≤ m, h i def ν ≤ 1. We consider the asymptotic rate defined as aR (ν, δ, ρ) = limm→∞ sup logqm2 AR (q m , n, d, r) .
parameters: ν =
ρ=
and δ =
We now investigate how AR (q m , n, d, r) behaves as the parameters tend to infinity. Without loss of generality, we only consider the case where 0 ≤ δ ≤ min{ν, 2ρ}, since aR (ν, δ, ρ) = 0 for δ > 2ρ. Proposition 17: For 0 ≤ δ ≤ ρ, aR (ν, δ, ρ) = ρ(1 + ν − ρ) − δ.
(28)
For ρ ≤ δ, we have to distinguish three cases. First, for 2ρ ≤ ν , (1 − ρ)(ν − ρ) ν−ρ max (2ρ − δ), (2ρ − δ), ρ(2ν − ρ) − νδ ≤ aR (ν, δ, ρ) ≤ (ν − ρ)(2ρ − δ). (29) 1 + 2ν − 3ρ 2 Second, for ν ≤ 2ρ ≤ 1, ρ(1 − ρ) ρ max (ν − δ), (2ν − 2ρ − δ), ρ(2ν − ρ) − νδ ≤ aR (ν, δ, ρ) ≤ ρ(ν − δ). 1+ρ 2 Third, for 2ρ ≥ 1, o nρ ρ (2 − 4ρ + ν − δ), (2ν − 2ρ − δ), ρ(2ν − ρ) − νδ, 0 ≤ aR (ν, δ, ρ) ≤ ρ(ν − δ). max 3 2 The proof of Proposition 17 is given in Appendix E.
(30)
(31)
Proposition 17 indicates that the codewords of rank r in an (n, n − d + 1, d) linear MRD code (d ≤ r ) form an asymptotically optimal (n, d, r) CRC. In particular, the set of codewords with rank n constitutes
17
a CRC of rank n and asymptotic rate of ν − δ, which is equal to the asymptotic rate of an optimal rank metric code [42]. The bounds on aR (ν, δ, ρ) given in Proposition 17 are illustrated in Figures 1, 2, and 3 for ν = 3/4 and ρ = 1/4, ρ = 2/5, and ρ = 3/5, respectively. Note that these three parameters correspond to the three cases in (29), (30), and (31), respectively.
0.4 δ≤ρ upper lower
0.35 0.3
aR(ν,δ,ρ)
0.25 0.2 0.15 0.1 0.05 0
Fig. 1.
0
0.1
0.2
δ
0.3
0.4
0.5
Asymptotic bounds on the maximal rate of a CRC as a function of δ, with ν = 3/4 and ρ = 1/4.
In Figures 1, 2, and 3, we can split the range of δ into two regions: when δ ≤ ρ, the asymptotic rate of CRCs is determined due to the construction of good CRCs when d ≤ r ; when δ ≥ ρ, we only have bounds on the asymptotic rate of CRCs. We remark that in (29), the first two lower bounds are competing, thus forming a triangle-shaped region for aR (ν, δ, ρ). However, for (30) and (31), the shape of the possible region for aR (ν, δ, ρ) depends on both ν and ρ. Also, the lower bounds based on the connection between CDCs and CRCs (the first two lower bounds in the LHS of (29), (30), and (31)) are tighter for 2ρ ≤ ν and on the other hand become trivial for ρ approaching 1.
18
0.7 δ≤ρ upper lower
0.6
aR(ν,δ,ρ)
0.5
0.4
0.3
0.2
0.1
0
Fig. 2.
0
0.1
0.2
0.3
0.4 δ
0.5
0.6
0.7
0.8
Asymptotic bounds on the maximal rate of a CRC as a function of δ, with ν = 3/4 and ρ = 2/5.
0.7 δ≤ρ upper lower
0.6
aR(ν,δ,ρ)
0.5
0.4
0.3
0.2
0.1
0
Fig. 3.
0
0.1
0.2
0.3
0.4 δ
0.5
0.6
0.7
0.8
Asymptotic bounds on the maximal rate of a CRC as a function of δ, with ν = 3/4 and ρ = 3/5.
19
A PPENDIX A. Proof of Proposition 4 Proof: Let us first focus on GF(q n ) and consider a basis Bn of GF(q n ) over GF(q). Define the def
code C over GF(q n ) formed by the codewords ci = bi Hi for 0 ≤ i ≤ |∆| − 1, where the expansion of bi ∈ GF(q n )r is given by HTi . Let us consider ci = bi Hi and cj = bj Hj . Let βi,k0 , βi,k1 , . . . , βi,kd−1 be
coordinates of bi linearly independent to the coordinates of bj and βj,l0 , βj,l1 , . . . , βj,ld−1 be coordinates of bj linearly independent to the coordinates of bi . We thus define the basis γi,j = {βi,k0 , βi,k1 , . . . , βi,kd−1 , βj,l0 , βj,l1 , . . . , βj,ld−1 , γ2d , γ2d+1 , . . . , γn−1 }
of GF(q n ) over GF(q). Expanding ci − cj with respect to the basis γi,j , we obtain rk(ci − cj ) ≥ ¯j ¯ i denotes the d rows of Hi corresponding to βi,k , βi,k , . . . , βi,k ¯ T , where H ¯T −H and H rk H d−1 1 0 j i denotes the d rows of Hj corresponding to βj,l0 , βj,l1 , . . . , βj,ld−1 . Thus
¯T −H ¯ T = rk(βi,k , βi,k , . . . , βi,k , −βj,l , −βj,l , . . . , −βj,l ) = 2d. rk H i j d−1 1 0 d−1 1 0
Using an argument similar to that in the proof of Prop. 3, we can show that dR ≤ d + r . For m ≥ n and a basis Bm of GF(q m ) over GF(q), we append m − n all-zero rows to the expansion with respect to Bn of codewords in C , then the matrices are the expansions (with respect to Bm ) of codewords of a CRC of length n and rank r over GF(q m ) and minimum rank distance ≥ 2d. We now derive the bounds on the diameter DR of C . Let x and y be distinct codewords in C such that dS (C(x), C(y)) = dS (R(x), R(y)) = DS , then by Theorem 1,
1 2
max{DS , 3DS − 2r} ≤ DR . Let x′
and y′ be distinct codewords in C such that dR (x′ , y′ ) = DR , then by Theorem 1, DR ≤ 12 (DS + 2r). B. Proof of Proposition 8 Proof: For all x ∈ GF(q m−1 )m−1 with rank m − 1, define g : x 7→ y ∈ GF(q m )m such that X 0 ∈ GF(q)m×m , Y= (32) 0 1 where X and Y are the expansions of x and y, respectively. By (32), for all x, x′ ∈ GF(q m−1 )m−1 with rank m − 1, we have rk(g(x)) = rk(x) + 1 = m and rk(g(x) − g(x′ )) = rk(x − x′ ). Therefore g is a homomorphism from Kq (m − 1, m − 1, d, m − 1) to Kq (m, m, d, m). Applying (6) to these graphs, and noticing that α(m, m) = q m−1 (q m − 1)α(m − 1, m − 1), we obtain (12).
20
We now prove (13). Note that any vector x ∈ GF(q m )n with rank r belongs to
n−r 1
ELSs of
dimension n − 1. Indeed, such ELSs are those which contain E , where E is the unique ELS of dimension r such that x ∈ E . Using basic counting, there are exactly n−r such ELSs. 1
Let C be an optimal (n, d, r) CRC over GF(q m ). For all c ∈ C and all V ∈ En−1 (q m , n), we define P f (V, c) = 1 if c ∈ V and f (V, c) = 0 otherwise. For all c, V∈En−1 (qm ,n) f (V, c) = n−r 1 , and for all P V , c∈C f (V, c) = |C ∩ V|. Summing over all possible pairs, we obtain X
V∈En−1
(q m ,n)
X
X
X
f (V, c) =
V∈En−1
c∈C
X
f (V, c) =
c∈C V∈En−1 (q m ,n)
|C ∩ V|
(q m ,n)
n−r AR (q m , n, d, r). 1
[n−r 1 ] A (q m , n, d, r). The [n1 ] R restriction of C ∩ U to the ELS U [35] is an (n − 1, d, r) CRC with cardinality |C ∩ U | over GF(q m ),
Hence there exists U ∈ En−1 (q m , n) such that |C ∩ U | =
and hence
q n−r −1 m q n −1 AR (q , n, d, r)
P
c∈C
f (U, c) ≥
≤ |C ∩ U | ≤ AR (q m , n − 1, d, r).
C. Proof of Proposition 12 Proof: For all x ∈ GF(q l )k with rank s and c ∈ C , we define fr (x, c) = 1 if dR (x, c) = r P l and fr (x, c) = 0 otherwise. Note that x:rk(x)=s fr (x, c) = J(q , k, s, r, rk(c)) for all c ∈ C and P l l k c∈C fr (x, c) = |{y ∈ C − x : rk(y) = r}| ≤ AR (q , k, d, r) for all x ∈ GF(q ) . We obtain X
X
n X
fr (x, c) =
X
(33)
i=0
c∈C x:rk(x)=s
X
Ai J(q l , k, s, r, i),
fr (x, c) ≤ Ns (q l , k)AR (q l , k, d, r).
(34)
x:rk(x)=s c∈C
Combining (33) and (34), we obtain m
AR (q , n, d, r) ≥
Pn
l i=0 Ai J(q , k, s, r, i) . Ns (q l , k)
(35)
Suppose d > r + 1. For all c ∈ C , let us denote the set of vectors with rank s at distance at most def S d − r − 1 from c as Sc , and S = c∈C Sc . For x ∈ Sc , we have dR (x, c) ≤ d − r − 1 < r . We have
for c′ ∈ C and c′ 6= c, dR (x, c′ ) ≥ dR (c, c′ ) − dR (x, c) ≥ r + 1; and hence fr (x, c′ ) = 0 for all c′ ∈ C . P Therefore, c∈C fr (x, c) = 0 for all x ∈ S and h i X X XX X X fr (x, c) ≤ Ns (q l , k) − |S| AR (q l , k, d, r). (36) fr (x, c) = fr (x, c) + x:rk(x)=s c∈C
x∈S c∈C
x∈S /
rk(x)=s
c∈C
21
Since d − r − 1 < d2 , the balls with radius d − r − 1 around the codewords are disjoint and hence P P |S| = ni=0 Ai d−r−1 J(q l , k, s, t, i). Combining (33) and (36), we obtain t=0 Pn l l i=0 Ai J(q , k, s, r, i) AR (q , k, d, r) ≥ . (37) P P n l , k, s, t, i) Ns (q l , k) − i=0 Ai d−r−1 J(q t=0 Note that (35) and (37) both hold for any s and weight spectrum {Ai }. Furthermore, since AR (q l , k, d, r)
is a non-decreasing function of l and k, AR (q m , n, d, r) ≥ AR (q l , k, d, r) for all max{r, d} ≤ k ≤ n and k ≤ l ≤ m. Thus, we have (18) and (19).
D. Proof of Proposition 15 Proof: By Proposition 11, we obtain AR (q m , m, d, m) ≤ α(m, m − d + 1) for r = n = m and AR (q m , n, d, r) ≤ nr α(m, r − d + 1) < nr q m(r−d+1) otherwise. We now derive lower bounds Pr j on M (q m , n, d, r). Again, M (q m , n, d, r) = nr j=d (−1) µj where µj > µj−1 for d + 1 ≤ j ≤ r . Therefore, when needed, we shall only consider the last terms in the summation. m
−1 m First, M (q m , m, m − 1, m) = (q 2m − 1) − qq−1 (q − 1) >
q−2 2m q−1 (q
− 1) >
q−2 q−1 α(m, 2),
which leads
to (24). For q = 2, M (2m , m, m − 1, m) = 2(2m − 1) = (2m−1 − 1)−1 α(m, 2), which results in (23). Second, when r = n = m and d = m − 2, α(m, 2) α(m, 1) 2m (q − 1) + 2 (q m − 1) q−1 (q − 1)(q − 1) q−2 1 α(m, 1)(q 2m − 1) + 2 α(m, 2)(q m − 1) q−1 (q − 1)(q − 1)
M (q m , m, m − 2, m) = (q 3m − 1) − > >
(q 2 − 1)(q − 2) + 1 α(m, 1), (q 2 − 1)(q − 1)
which leads to (25). Third, when r = n = m and d < m − 2, by considering the last four terms in the summation, we obtain M (q m , m, d, m) > (q m(m−d+1) − 1) −
α(m, 1) m(m−d) (q − 1) q−1
α(m, 2) α(m, 3) (q m(m−d−1) − 1) − 3 (q m(m−d−2) − 1) (q 2 − 1)(q − 1) (q − 1)(q 2 − 1)(q − 1) q3 − 2 q−2 + α(m, m − d + 1), > q − 1 (q 3 − 1)(q 2 − 1)(q − 1)
+
which results in (26).
22
Fourth, when d < r < m, by considering the last two terms in the summation, we obtain n r m(r−d) M (q m , n, d, r) ≥ (q m(r−d+1) − 1) − (q − 1) r 1 n q m(r−d+1) − 1 − q m(r−d)+r + q r ≥ r n m(r−d+1) ≥ q (1 − q r−m ). r Therefore, since r < m, B(q m , n, d, r) < (1 − q r−m )−1 ≤
q q−1 ,
(38) (39)
which leads to (27).
E. Proof of Proposition 17 Proof: We first derive a lower bound on aR (ν, δ, ρ). We shall use the following bounds on the def Q∞ −j Gaussian polynomial: q r(n−r) ≤ nr < Kq−1 q r(n−r) , where Kq = j=1 (1 − q ) [35, Lemma 1].
For d ≤ r , Proposition 14 yields AR (q m , n, d, r) ≥ q r(n−r)+m(r−d) , which asymptotically becomes aR (ν, δ, ρ) ≥ ρ(1 + ν − ρ) − δ for δ ≤ ρ. Similarly, for d > r , Proposition 16 yields AR (q m , n, d, r) ≥ q r(n−r)+n(r−d+1) , which asymptotically becomes aR (ν, δ, ρ) ≥ ρ(2ν − ρ) − νδ for δ ≥ ρ. Also, by (5),
(9) asymptotically yields aR (ν, δ, ρ) ≥ min{(ν − ρ)(ρ − 2δ ), ρ(ν − ρ − 2δ )} for δ ≥ ρ. Proposition 5 and (5) yield logq AR (q m , n, d, r) ≥ min{(n − r)(2r − d − 2p + 1), (m − r)(p + 1)} for d > r and 2r ≤ n. Treating the two terms as functions and assuming that p is real, the lower bound is
maximized when p=
Using p =
j
(n−r)(2r−d+1)−m+r m+2n−3r
k
(n − r)(2r − d + 1) − m + r . m + 2n − 3r
, asymptotically we obtain aR (ν, δ, ρ) ≥
(40) (1−ρ)(ν−ρ) 1+2ν−3ρ (2ρ
− δ) for 2ρ ≤ ν .
For d > r and n ≤ 2r ≤ m, Proposition 5 and (5) lead to logq AR (q m , n, d, r) ≥ min{r(n − d − 2p + 1), (m − r)(p + 1)}. After maximizing this expression over p, we asymptotically obtain aR (ν, δ, ρ) ≥ ρ(1−ρ) 1+ρ (ν
− δ) for ν ≤ 2ρ ≤ 1.
For d > r and 2r ≥ m, Proposition 5 and (5) lead to logq AR (q m , n, d, r) ≥ min{r(n − d − 2p + 1), r(m − 2r + p + 1)}. After maximizing this expression over p, we asymptotically obtain aR (ν, δ, ρ) ≥ ρ 3 (2
− 4ρ + ν − δ) for 2ρ ≥ 1.
We now derive an upper bound on aR (ν, δ, ρ). First, Proposition 11 gives AR (q m , n, d, r)
