DIFFERENCE SETS AND CODES IN An LATTICES
DIFFERENCE SETS AND CODES IN An LATTICES
arXiv:1409.5276v2 [math.CO] 30 Oct 2014
ˇ ´ MLADEN KOVACEVI C Abstract. This paper presents a geometric/coding theoretic interpretation of planar difference sets and the corresponding reformulations of two important conjectures in the field. It is shown that such sets of order n exist if and only if the An lattice admits a lattice tiling by balls of radius 2 under ℓ1 metric (i.e., a linear 2-perfect code). More general difference sets and perfect codes of larger radius are also studied. Several communication scenarios are described for which the obtained results, and codes in An lattices in general, are relevant.
1. Preliminaries 1.1. Difference sets. Let G be a group of order v (written additively). A set D ⊆ G of size k is said to be a (v, k, λ)-difference set if every nonzero element of G can be expressed as a difference d1 − d2 of two elements from D in exactly λ ways. The parameters v, k, λ then necessarily satisfy the identity λ(v − 1) = k(k − 1). The order of such a difference set is defined as n = k − λ. If G is Abelian, cyclic, etc., then D is also said to be Abelian, cyclic, etc., respectively. We shall usually omit the word Abelian in the sequel, as this is the only case that will be treated. Furthermore, we shall mostly be concerned with planar (or simple) difference sets – those with λ = 1. Difference sets are very well-studied, and a large body of literature is devoted to the investigation of their properties1. Some of the most interesting problems regarding difference sets concern the existence of these objects for specific sets of parameters. One of the most familiar such problems, known as the prime power conjecture [4, Conj. 7.5, p. 346], is the following: Planar difference set of order n exists if and only if n is a prime power (counting n = 1 as a prime power). Existence of such sets for n = pm , p prime, m ∈ Z≥0 , was demonstrated by Singer [26], but the necessity of this condition remains an open problem for more than seven decades. In this note we intend to present a geometric interpretation of difference sets, offering a different view on the subject and potentially another approach to studying them. 1.2. An lattice under ℓ1 metric. The An lattice is defined as ) ( n X xi = 0 (1.1) An = (x0 , x1 , . . . , xn ) : xi ∈ Z, i=0
where Z denotes the integers, as usual. A1 is equivalent to Z, A2 to the hexagonal lattice, and A3 to the face-centered cubic lattice (see [8]). The metric on An that we understand Date: October 31, 2014. 2010 Mathematics Subject Classification. Primary: 05B10, 05B45, 94B25; Secondary: 52C22, 52C07, 68P30, 68R05. Key words and phrases. Difference set, prime power conjecture, An lattice, perfect code, tiling, group splitting, dominating set, covering code. This work was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia (grants TR32040 and III44003), and by the EU COST action IC1104. 1For a nice account of the theory and an extensive list of references see [4], and also [17, 18, 19]. For some of the applications of difference sets in communications and coding theory see, e.g., [1, 23, 31, 7, 10]. 1
ˇ ´ MLADEN KOVACEVI C
2
is a variation of the ℓ1 distance n
1X 1 |xi − yi |, d(x, y) = kx − yk1 = 2 2
(1.2)
i=0
where x = (x0 , x1 , . . . , xn ), y = (y0 , y1 , . . . , yn ). The constant 1/2 is taken for convenience because kx − yk1 is always even for x, y ∈ An . Distance d also represents the graph distance in An ; namely, if Γ(An ) is a graph with the vertex set An and with edges joining neighboring points (i.e., points at distance 1 under d), then d(x, y) is the length of the shortest path between x and y in Γ(An ). 2 Ball of radius 1 around x ∈ An contains 2 n+1 2 +1 = n +n+1 points of the form x+fi,j , where fi,j is a permutation of (1, −1, 0, . . . , 0) having a 1 at the i’th coordinate, a −1 at the j’th coordinate, and zeros elsewhere (with the convention fi,i = 0). In two dimensions we can visualize this ball as a hexagon, and in three as a so-called cuboctahedron (Fig. 1). In general, convex interior of the points in the ball forms a highly symmetrical polytope having the following interesting property, among many others – the distance of all the vertices from the center is equal to the distance between any two neighboring vertices.
Figure 1. Balls of radius 1 in (A2 , d) – hexagon, and in (A3 , d) – cuboctahedron. Ball of radius r around x ∈ An contains all the points with integral coordinates in the convex interior of {x + rfi,j }. For studying packing problems, it is usually simpler to visualize Zn instead of an arbitrary lattice. In our case there is a trivial map that makes the transition to Zn and back very easy. For x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ Zn , define the metric X X (1.3) d+ (x, y) = max (xi − yi ), (yi − xi ) . i: xi >yi
i: xi yi
because
(1.5)
Pn
i=0 xi
=
n X
(yi − xi )
i=0 xi yi ′
xi 0, xj1 , . . . , xjn−k ≤ 0, where k ∈ {0, . . . , n} and {i1 , . . . , ik , j1 , . . . , jn−k } = {1, . . . , n}. It is clear that the number of (k) points in one subset depends only on k; denoting it by Mn (r), we can write Mn (r) = P n (k) k Mn (r). Observe next that the “positive mass” is distributed on the coordinates xi1 , . . . , xik independently of the way the “negative mass” is distributed over xj1 , . . . , xjn−k , (k)
and hence Mn (r) is equal to the product of the number of vectors (xi1 , . . . , xik ) ∈ Zk with P xiu > 0 and ku=1 xiu ≤ r, and the number of vectors (xj1 , . . . , xjn−k ) ∈ Zn−k with xjv ≤ 0 Pn−k and v=1 xjv ≥ −r. The first number is precisely kr , and the second r+n−k n−k . 2. Difference sets as sublattices of An
In the following, when using concepts from graph theory in our setting, we have in mind the graph representation Γ(An ) of An , as introduced above. An (r, i, j)-cover (or (r, i, j)covering code) in a graph Γ = (V, E) [2] is a set of its vertices S ⊆ V with the property that every element of S is covered by exactly i balls of radius r centered at elements of S, while every element of V \ S is covered by exactly j such balls. Special cases of such sets, namely (1, i, j)–covers, have also been studied in the context of domination theory in graphs [30]. In coding theory, (r, 1, 1)-covers are known as r-perfect codes. An independent set in a graph Γ = (V, E) is a subset of its vertices I ⊆ V , no two of which are adjacent in Γ. 2.1. The general case. The proof of the following theorem uses the same method that was employed to prove the connection between lattice tilings and group splitting [14, 28] (see also [29]). In fact, planar difference sets (λ = 1) are an instance of splitting sequences. Theorem 2.1. A (v, k, λ)-difference set exists if and only if the lattice Ak−1 contains a (1, 1, λ)-covering sublattice. Proof. Suppose that D = {d0 , d1 , . . . , dk−1 } is a (v, k, λ)-difference set in a group G. Observe the sublattice ) ( k−1 X xi di = 0 (2.1) L = x ∈ Ak−1 : i=0
where xi di denotes the sum (in G) of |xi | copies of di if xi > 0, and of −di if xi < 0. L is a (1, 1, in An . To see this, consider a point y = (y0 , y1 , . . . , yk−1 ) ∈ / L, meaning Pλ)-cover n that i=0 yi di = a ∈ G, a 6= 0. The neighbors of y are of the form y + fi,j , i 6= j. Since D is a difference set, we know that −a ∈ G can be written as a difference of the elements from D in exactly λ different ways, meaning that there are λ different pairs (s, t) for which ds − dt = −a, dsP , dt ∈ D. ForPevery such pair observe the point zs,t = y + fs,t . We have zs,t ∈ L because ni=0 zi di = ni=0 yi di + ds − dt = a − a = 0. Therefore, there are exactly λ points in the lattice L that are adjacent to y, i.e., such that balls of radius 2This fact is mentioned in [11] for the case n = 2, though the interpretation via the metric d is not +
given.
4
ˇ ´ MLADEN KOVACEVI C
1 around them cover y. To show that the elements of L are covered only by the balls around themselves (i.e., that L is an independent set in Γ(An )), observe that if there were two points at distance 1 in L, then by the same argument as above we would obtain that ds − dt = 0, i.e., ds = dt for some s 6= t, which is a contradiction because |D| = k. For the other direction, assume that L′ is a (1, 1, λ)-covering sublattice of Ak−1 . Define the group G = Ak−1 /L′ , and observe D = {d0 , d1 , . . . , dk−1 } ⊆ G, where di = [fi,0 ] ≡ fi,0 + L′ are the cosets (elements of G). Let us first assure that all the di ’s are distinct. Suppose that ds = dt for some s 6= t. This implies that ds − dt = [fs,t ] = [0], which means that fs,t ∈ L′ . But since 0 ∈ L′ , and 0 and fs,t are at distance 1, this would contradict the fact that L′ is independent. Hence, |D| = k. Now take any nonzero element of G, say [y], y ∈ / L′ . By assumption, y is covered by exactly λ elements of L′ , i.e., y + fs,t ∈ L′ for exactly λ vectors fs,t. Since fs,t = fs,0 − ft,0 , this is equivalent to saying that dt − ds = [ft,0 ] − [fs,0 ] = [y]. Therefore, every nonzero element of G can be written as a difference of two elements from D in exactly λ different ways. Geometrically, the theorem states that the balls of radius 1 around the points of the sublattice L overlap in such a way that every point that does not belong to L is covered by exactly λ balls. (The elements of L, i.e., centers of the balls, are covered by one ball only, and hence this notion is different than that of multitiling [13].) Example 1. D = {0, 1, 2} is a (4, 3, 2)-difference set in the cyclic group Z4 (integers modulo 4). A (1, 1, 2)-covering sublattice L ⊂ A2 corresponding to this difference set is illustrated in Fig. 2. Points in L are depicted as black, and those in A2 \ L as white dots. For illustration, Fig. 3 shows an example of a (1, 2, 2)-covering sublattice (which does not correspond to any difference set). △
Figure 2. A (1, 1, 2)-covering sublattice of A2 . Remark 2.2. We can interpret the lattice L as an error-correcting/detecting code, in which case the points in L are called codewords. When λ = 1, the code can correct a single error because the balls of radius one around codewords do not overlap and the minimum distance of the code is three (here by a single error we mean the addition of a vector fi,j for some i, j, i 6= j, to the “transmitted” codeword x ∈ L). For λ > 1, however, it can only detect a single error reliably. Note also that increasing λ increases the density of the code/lattice L in Ak−1 , but does not affect its error-detection capability. The densest such lattice is therefore obtained for λ = k (that this is the maximum value follows from λ(v − 1) = k(k − 1) and k ≤ v); it corresponds to the trivial (v, v, v)-difference set D = G in an arbitrary group G. △
DIFFERENCE SETS AND CODES IN An LATTICES
5
Figure 3. A (1, 2, 2)-covering sublattice of A2 . Note that we have not specified the order of the elements of the difference set D when defining the corresponding lattice L in (2.1). Therefore, more than one lattice can be defined via D, but they are all identical up to a permutation of the coordinates. Note also that if we write d′i = zdi + g instead of di in (2.1), where z is a fixed integer coprime with v and g is a fixed element of G, identical lattice is obtained because k−1 X
(2.2)
i=0
xi di = 0
⇔
k−1 X
xi d′i = 0
i=0
Pk−1
which follows from i=0 xi = 0 and gcd(z, v) = 1. Let us recall some terminology. Two difference sets D and D ′ in an Abelian group G are said to be equivalent [4, Rem. 1.11, p. 302] if D ′ = {zd + g : d ∈ D}, for some z ∈ Z coprime with v and some g ∈ G. Two codes C and C ′ of length m over an alphabet A are equivalent [24, p. 40] if there exist m permutations of A, π1 , . . . , πm , and a permutation σ over {1, . . . , m} such that (2.3) C ′ = σ(π1 (x1 ), . . . , πm (xm )) : (x1 , . . . , xm ) ∈ C . We then have the following:
Proposition 2.3. If two difference sets D and D ′ are equivalent, then the corresponding codes (defined as in (2.1)) are equivalent. In fact, the πi ’s are necessarily identity maps, only σ is relevant here. 2.2. Planar difference sets as perfect codes. An interesting special case of Theorem 2.1 is obtained by considering planar difference sets (λ = 1). Recall that an r-perfect code in a discrete metric space (U, d) is a subset C ⊆ U with the property that balls of radius r around the “codewords” from C are disjoint and cover the entire space U . In the above terminology r-perfect codes are (r, 1, 1)-covers; 1-perfect codes are also called efficient dominating sets in graph theory [3]. When U is a vector space and C its subspace, C is said to be a linear code. The same terminology is often used for lattices, namely, C is called a linear code if it is a sublattice of the lattice in question. Corollary 2.4. Planar difference set of order n exists if and only if the space (An , d) admits a linear 1-perfect code (i.e., an efficient dominating sublattice).
6
ˇ ´ MLADEN KOVACEVI C
Existence of such codes when n is a prime power follows from the existence of the corresponding difference sets [26], but the necessity of this condition is open and is equivalent to the prime power conjecture. Conjecture 2.5 (Prime power conjecture). There exists a linear 1-perfect code in (An , d) (or, equivalently, in (Zn , d+ )) if and only if n is a prime power. △ A stronger conjecture would claim the above even for nonlinear codes. Example 2. Consider the (13, 4, 1)-difference set D = {0, 1, 3, 9} ⊂ Z13 . The corresponding 1-perfect code in (A3 , d) is illustrated in Fig. 4. The figure shows the intersection of A3 with the plane x0 = 0; the intersections of a ball of radius 1 in A3 with the planes x0 = const are shown in Fig. 5 for clarification. △
Figure 4. 1-perfect code in (A3 , d).
Figure 5. Intersections of a ball in (A3 , d) with the planes x0 = const. Another important unsolved conjecture in the field is the following: All Abelian planar difference sets live in cyclic groups [4, Conj. 7.7, p. 346]. Since the group G containing the difference set which defines the code L is isomorphic to An /L (see the proof of Theorem 2.1), the statement that G is cyclic, i.e., that it has a generator, is equivalent to the following: Conjecture 2.6 (All Abelian planar difference sets are cyclic). Let L be a 1-perfect code in (An , d). Then the period of L in An along the direction fi,j is equal to n2 + n + 1 for at least one vector fi,j , (i, j) ∈ {0, 1, . . . , n}2 . △
DIFFERENCE SETS AND CODES IN An LATTICES
7
The cyclic case. In the rest of this subsection we restrict our attention to cyclic planar difference sets of order n, i.e., it is assumed that the group we are working with is Zv , v = n2 + n + 1; as mentioned above, this in fact might not be a restriction at all. So let D = {d0 , d1 , . . . , dn } ⊂ Zv be a difference set and L the corresponding code (see (2.1)). We will assume that d0 = 0, d1 = 1. (This is not a loss in generality because if D is a difference set, there exist two elements, say d0 , d1 ∈ D, such that d1 − d0 = 1. Then we can take the equivalent difference set D ′ = {di − d0 : di ∈ D} which obviously contains 0 and 1.) In this case the generator matrix of the code (lattice) L has the following form v 0 0 ··· 0 −d2 1 0 · · · 0 (2.4) B(L) = −d3 0 1 · · · 0 , .. .. .. .. . . . . . . . −dn 0 0 · · · 1
i.e., the codewords are the vectors x = ξ · B(L), ξ ∈ Zn (the The generator matrix of the dual lattice L∗ is 1 d d3 2 ··· v v v 0 1 0 · · · ∗ −1 (2.5) B(L ) = B(L) = 0 0 1 · · · .. .. .. . . . . . . 0 0 0 ···
vectors are written as rows). dn v
0 0 . .. . 1
We have disregarded above the 0-coordinate because d0 = 0. Therefore, B(L) is in fact a generator matrix of the corresponding code in (Zn , d+ ) (see Lemma 1.1).
Finite alphabet. By taking the codewords of L modulo v = n2 + n + 1, one obtains a finite code in Znv defined by the generator matrix (over Zv ) −d2 1 0 · · · 0 −d3 0 1 · · · 0 (2.6) .. .. .. . . .. . . . . . . −dn 0 0 · · ·
1
This code is of length n, has v n−1 codewords, and is 1-perfect (under the obvious “modulo v version” of the d+ metric). It is also systematic, i.e., the information sequence itself is a part of the codeword. The “parity check” matrix of the code is H = 1 d2 · · · dn . Thus, the codewords are all those vectors x = (x1 , . . . , xn ) ∈ Znv for which H ·xT = 0 mod v, and the syndromes T = d −d . of the correctable error vectors fi,j are H · fi,j i j 3. r-perfect codes in (An , d) A natural question related to the results of Section 2.2 is whether there exist r-perfect codes in (An , d) for r > 1. For example, such codes exist for any r in dimensions one and two: it is easy to verify that the sublattice of A1 spanned by the vector (−2r − 1, 2r + 1) is an r-perfect code in A1 , and the sublattice of A2 spanned by the vectors (−2r − 1, r, r + 1), (−r − 1, 2r + 1, r) is an r-perfect code in A2 (for a study of the two-dimensional case see also [9]). In higher dimensions, however, it does not seem to be possible to tile (An , d) with balls of radius r > 1. We shall not be able to prove this claim here, but Theorem 3.2 below is a step in this direction. Denote by Bn (x, r) = {y ∈ Zn : d+ (x, y) ≤ r} the ball of radius r around x in (Zn , d+ ). Let Dn (r) be the body in Rn defined as the union of unit cubes translated to the points
8
ˇ ´ MLADEN KOVACEVI C
Figure 6. Bodies in R2 corresponding to a ball of radius 2 in Z2 , d+ : The cubical tile (left) and the convex interior (right). S of Bn (0, r), namely, Dn (r) = y∈Bn (0,r) (y + [−1/2, 1/2]n ) (see Fig. 6). Let also Cn (r) be the body defined as the convex interior (in Rn ) of the points in Bn (0, r). Lemma 3.1. The volumes of the bodies Dn (r) and Cn (r) are given by (3.1) (3.2)
Vol(Dn (r)) = Vol(Cn (r)) =
min{n,r}
X
k=0 n r 2n
n!
n
n k
r r+n−k k n−k
.
Furthermore, limr→∞ Vol(Cn (r))/ Vol(Dn (r)) = 1. Proof. Since Dn (r) consists of unit cubes, its volume is Vol(Dn (r)) = |Bn (x, r)|, which gives the above expression by Lemma 1.2. The volume of Cn (r) can be computed by the method similar to the proof of Lemma 1.2, namely, we observe the intersection of Cn (r) with the orthant x1 , . . . , xk > 0, xk+1 , . . . , xn ≤ 0, where k ∈ {0, of this intersection is the product of the volumes of . . . , n}. The volume P the k-simplex (x , . . . , x ) : x > 0, x ≤ r , which is known to be r k /k!, and of the i i k 1 P (n − k)-simplex (xk+1 , . . . , xn ) : xi ≤ 0, xi ≥ −r , which is r n−k /(n − k)!. Therefore, k rn−k 2 P P Vol(Cn (r)) = nk=0 nk rk! (n−k)! , which, together with the identity nk=0 nk = 2n n , gives (3.2). The asymptotic behavior (when r → ∞) of Vol(Dn (r)) is easily found to be Vol(Dn (r)) ∼ r n 2n by using the fact that kr ∼ r k /k!. n! n Theorem 3.2. There are no r-perfect codes in (An , d), n ≥ 3, for large enough r, i.e., for r ≥ r0 (n).
Proof. The proof is based on the same idea as the one for r-perfect codes in Zn under ℓ1 (also termed Manhattan or taxi) distance [12]. First observe that an r-perfect code in (Zn , d+ ) would induce a tiling of Rn with Dn (r), and a packing with Cn (r). The relative efficiency of the latter with respect to the former is defined as the ratio of the volumes of these bodies, Vol(Cn (r))/ Vol(Dn (r)), which by Lemma 3.1 tends to 1 as r grows indefinitely. This has the following consequence: If an r-perfect code exists in (Zn , d+ ) for arbitrarily large r, then there exists a tiling of Rn by translates of the body Dn (r) for arbitrarily large r, which further implies that a packing of Rn by translates of the body Cn (r) exists which has efficiency arbitrarily close to 1. But then there would also be a packing by Cn (r) of efficiency 1, i.e., a tiling (in [12, Appendix] it is shown that there exists a packing whose density is the supremum of the densities of all possible packings with a given body). This is a contradiction. Namely, by [25, Thm 1] a necessary condition
DIFFERENCE SETS AND CODES IN An LATTICES
9
for a convex body to be able to tile space is that it be a centrally symmetric polytope3 with centrally symmetric facets, which the polytope Cn (r) fails to satisfy for n ≥ 3. For example,the facet which is thePintersection of Cn (r) with the hyperplane x1 = −r is the simplex (x2 , . . . , xn ) : xi ≥ 0, ni=2 xi ≤ r , a non-centrally-symmetric body. In summary, we have shown that linear r-perfect codes in (An , d) exist for: • n = 1, r arbitrary, • n = 2, r arbitrary, • n ≥ 3 a prime power, r = 1. The statement that these are the only cases (apart from the trivial one r = 0), even if nonlinear codes are allowed, is a further strengthening of the prime power conjecture. It should also be contrasted with the Golomb-Welch conjecture [12] (see also, e.g., [15, 16]) stating that r-perfect codes in Zn under ℓ1 distance exist only in the following cases: 1) n ∈ {1, 2}, r arbitrary, and 2) r = 1, n arbitrary. 4. Applications in coding theory 4.1. Permutation channels. A permutation channel [21, 22] over an alphabet A is a communication channel that takes sequences of symbols from A as inputs, and for any input sequence outputs a random permutation of this sequence. This channel is intended to model packet networks based on routing in which the receiver cannot rely on the packets being delivered in any particular order, as well as several other communication scenarios where a similar effect occurs, such as systems for distributed storage, data gathering in wireless sensor networks, etc. It was shown in [22] that the appropriate space in which error-correcting codes for such channels should be defined is (∆nℓ , d), where ) ( n X n+1 n xi = ℓ (4.1) ∆ℓ = (x0 , x1 , . . . , xn ) ∈ Z≥0 : i=0
is the discrete standard simplex, and d is the metric given by (1.2). Notice that ∆nℓ is just the translated An lattice restricted to the nonnegative orthant. This restriction is the reason that this space lacks some nice properties that are usually exploited when studying bounds on codes, packing problems, and the like. In order to study the underlying geometric problem, one can disregard these restrictions and investigate the corresponding problems in (An , d). The same approach is employed for some other types of codes; for example, studying the geometry of codes for flash memories reduces to packing problems in Zn , see, e.g., [5, 6, 27]. 4.2. Particle insertion/deletion channels. Consider the following channel model. The transmitter sends xi particles (or packets) in the i’th time slot. The particles are assumed identical, implying that the transmitted sequence can be identified with a sequence of nonnegative integers (x1 , . . . , xn ) ∈ Zn≥0 . In the channel, some of the particles can be lost (deletions), while new particles can appear (insertions) from the surrounding medium. Additionally, one can also allow delays of the particles in the model [20]. (Notice that if a particle is delayed, this can be thought of as it being deleted, while another particle is being inserted several time slots later.) Such channels are of interest in molecular communications, discrete-time queuing systems, etc. We want to guarantee that all patterns of ≤ r insertions and ≤ r deletions can be corrected at the receiving side. In other words, we want to be able to recover from all 3A polytope P ⊂ Rn is centrally symmetric if its translation P˜ = P − x satisfies P˜ = −P˜ for some
x ∈ Rn .
10
ˇ ´ MLADEN KOVACEVI C
patterns of errors such that4 max{#deletions, #insertions} ≤ r. It is not difficult to see that this will be achieved if and only if the d+ -balls of radius r around codewords are disjoint. In other words, the decoding regions are in this case defined precisely by balls in d+ metric. Hence, packing/tiling problems in (Zn , d+ ) are indeed relevant for studying and designing good codes for channels of this type. 4.3. Channels with ordered alphabets. Observe the channel with input and output alphabet A = {0, 1, . . . , q − 1} and think of the symbols in A as voltage levels (e.g., in baseband digital signal transmission, or in multi-level flash memory [6, 27]). Assume that we wish to assure that the receiver will be able to recover the signal whenever the total voltage drop (across all symbols of a particular codeword) and the total voltage increase caused by noise are at most r. This situation is essentially identical to the one in the previous example, and it is clear that the codes in (Zn , d+ ) (more precisely their restrictions to An ) are adequate constructions for error-correction in this scenario. Acknowledgment The author would like to thank Dr Moshe Schwartz for reading the manuscript and providing several useful comments. References 1. M. D. Atkinson, N. Santoro, and J. Urrutia, “Integer Sets with Distinct Sums and Differences and Carrier Frequency Assignments for Nonlinear Repeaters,” IEEE Trans. Commun., vol. 34, no. 6, pp. 614–617, Jun. 1986. 2. M. A. Axenovich, “On Multiple Coverings of the Infinite Rectangular Grid with Balls of Constant Radius,” Discrete Math., vol. 268, no. 1–3, pp. 31–48, Jul. 2003. 3. D. W. Bange, A. E. Barkauskas, and P. J. Slater, “Efficient Dominating Sets in Graphs,” in: R. D. Ringeisen and F. S. Roberts (eds.), Applications of Discrete Mathematics, SIAM, Philadelphia, 1988, pp. 189–199. 4. T. Beth, D. Jungnickel, and H. Lenz, Design Theory, 2nd ed., Cambridge University Press, 1999. 5. S. Buzaglo and T. Etzion, “Tilings with n-Dimensional Chairs and Their Applications to Asymmetric Codes,” IEEE Trans. Inform. Theory, vol. 59, no. 3, pp. 1573–1582, Mar. 2013. 6. Y. Cassuto, M. Schwartz, V. Bohossian, and J. Bruck, “Codes for Asymmetric Limited-Magnitude Errors with Application to Multilevel Flash Memories,” IEEE Trans. Inform. Theory, vol. 56, no. 4, pp. 1582–1595, Apr. 2010. 7. C. Chen, B. Bai, and X. Wang, “Construction of Nonbinary Quasi-Cyclic LDPC Cycle Codes Based on Singer Perfect Difference Set,” IEEE Commun. Lett., vol. 14, no. 2, pp. 181–183, Feb. 2010. 8. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Springer, 1999. 9. S. I. R. Costa, M. Muniz, E. Agustini, and R. Palazzo, “Graphs, Tessellations, and Perfect Codes on Flat Tori,” IEEE Trans. Inform. Theory, vol. 50, no. 10, pp. 2363–2377, Oct. 2004. 10. M. Esmaeili and M. Javedankherad, “4-Cycle Free LDPC Codes Based on Difference Sets,” IEEE Trans. Commun., vol. 60, no. 12, pp. 3579–3586, Dec. 2012. 11. T. Etzion, “Sequence Folding, Lattice Tiling, and Multidimensional Coding” IEEE Trans. Inform. Theory, vol. 57, no. 7, pp. 4383–4400, Jul. 2011. 12. S. W. Golomb and L. R. Welch, “Perfect Codes in the Lee Metric and the Packing of Polyominoes,” SIAM J. Appl. Math., vol. 18, no. 2, pp. 302–317, Mar. 1970. 13. N. Gravin, S. Robins, and D. Shiryaev, “Translational Tilings by a Polytope, with Multiplicity,” Combinatorica, vol. 32, no. 6, pp. 629–648, Dec. 2012. 14. W. Hamaker, “Factoring Groups and Tiling Space,” Aequationes Math., vol. 9, no. 2-3, pp. 145–149, 1973. 15. P. Horak, “Tilings in Lee Metric,” European J. Combin., vol. 30, no. 2, pp. 480–489, Feb. 2009. 16. P. Horak, “On Perfect Lee Codes,” Discrete Math., vol. 309, no. 18, pp. 5551–5561, Sep. 2009. 17. D. Jungnickel, “Difference Sets,” in: Contemporary Design Theory: A Collection of Surveys (Eds. J. H. Dinitz and D. R. Stinson), Wiley, New York, 1992, pp. 241–324. 4Notice that this approach is slightly different from the standard one where one requires that the
patterns of errors with (#deletions + #insertions) ≤ r are correctable. Thus, we treat deletions and insertions separately here because these impairments are of different nature and different origin.
DIFFERENCE SETS AND CODES IN An LATTICES
11
18. D. Jungnickel and B. Schmidt, “Difference Sets: An Update,” in: Geometry, Combinatorial Designs and Related Structures (Eds. J. W. P. Hirschfeld, S. S. Magliveras, and M. J. de Resmini), pp. 89–112, Cambridge University Press, 1997. 19. D. Jungnickel and B. Schmidt, “Difference Sets: A Second Update,” Rend. Circ. Mat. Palermo Serie 2 Suppl., vol. 53, pp. 89–118, 1998. 20. M. Kovaˇcevi´c and P. Popovski, “Zero-Error Capacity of a Class of Timing Channels,” IEEE Trans. Inform. Theory, vol. 60, no. 11, pp. 6796–6800, Nov. 2014. 21. M. Kovaˇcevi´c and D. Vukobratovi´c, “Multiset Codes for Permutation Channels,” preprint available at arXiv:1301.7564. 22. M. Kovaˇcevi´c and D. Vukobratovi´c, “Perfect Codes in the Discrete Simplex,” Des. Codes Cryptogr., to appear, DOI: 10.1007/s10623-013-9893-5. 23. A. W. Lam and D. V. Sarwate, “On Optimum Time-Hopping Patterns,” IEEE Trans. Commun., vol. 36, no. 3, pp, 380–382, Mar. 1988. 24. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company, 1977. 25. P. McMullen, “Convex Bodies Which Tile Space by Translation,” Mathematika, vol. 27, no. 1, pp. 113–121, Jun. 1980. 26. J. Singer, “A Theorem in Finite Projective Geometry and Some Applications to Number Theory,” Trans. Amer. Math. Soc., vol. 43, pp. 377–385, 1938. 27. M. Schwartz, “Quasi-Cross Lattice Tilings with Applications to Flash Memory,” IEEE Trans. Inform. Theory, vol. 58, no. 4, pp. 2397–2405, Apr. 2012. 28. S. Stein, “Packings of Rn by Certain Error Spheres,” IEEE Trans. Inform. Theory, vol. 30, no. 2, pp. 356–363, Mar. 1984. 29. S. Stein and S. Szab´ o, Algebra and Tiling: Homomorphisms in the Service of Geometry, The Mathematical Association of America, 1994. 30. J. A. Telle, “Complexity of Domination-Type Problems in Graphs,” Nordic J. Comput., vol. 1, pp. 157–171, 1994. 31. E. J. Weldon, Jr., “Difference-Set Cyclic Codes,” Bell Syst. Tech. J., vol. 45, pp. 1045–1055, 1966. ´a Department of Electrical Engineering, University of Novi Sad, Trg Dositeja Obradovic 6, 21000 Novi Sad, Serbia. E-mail address: [email protected]
arXiv:1409.5276v2 [math.CO] 30 Oct 2014
ˇ ´ MLADEN KOVACEVI C Abstract. This paper presents a geometric/coding theoretic interpretation of planar difference sets and the corresponding reformulations of two important conjectures in the field. It is shown that such sets of order n exist if and only if the An lattice admits a lattice tiling by balls of radius 2 under ℓ1 metric (i.e., a linear 2-perfect code). More general difference sets and perfect codes of larger radius are also studied. Several communication scenarios are described for which the obtained results, and codes in An lattices in general, are relevant.
1. Preliminaries 1.1. Difference sets. Let G be a group of order v (written additively). A set D ⊆ G of size k is said to be a (v, k, λ)-difference set if every nonzero element of G can be expressed as a difference d1 − d2 of two elements from D in exactly λ ways. The parameters v, k, λ then necessarily satisfy the identity λ(v − 1) = k(k − 1). The order of such a difference set is defined as n = k − λ. If G is Abelian, cyclic, etc., then D is also said to be Abelian, cyclic, etc., respectively. We shall usually omit the word Abelian in the sequel, as this is the only case that will be treated. Furthermore, we shall mostly be concerned with planar (or simple) difference sets – those with λ = 1. Difference sets are very well-studied, and a large body of literature is devoted to the investigation of their properties1. Some of the most interesting problems regarding difference sets concern the existence of these objects for specific sets of parameters. One of the most familiar such problems, known as the prime power conjecture [4, Conj. 7.5, p. 346], is the following: Planar difference set of order n exists if and only if n is a prime power (counting n = 1 as a prime power). Existence of such sets for n = pm , p prime, m ∈ Z≥0 , was demonstrated by Singer [26], but the necessity of this condition remains an open problem for more than seven decades. In this note we intend to present a geometric interpretation of difference sets, offering a different view on the subject and potentially another approach to studying them. 1.2. An lattice under ℓ1 metric. The An lattice is defined as ) ( n X xi = 0 (1.1) An = (x0 , x1 , . . . , xn ) : xi ∈ Z, i=0
where Z denotes the integers, as usual. A1 is equivalent to Z, A2 to the hexagonal lattice, and A3 to the face-centered cubic lattice (see [8]). The metric on An that we understand Date: October 31, 2014. 2010 Mathematics Subject Classification. Primary: 05B10, 05B45, 94B25; Secondary: 52C22, 52C07, 68P30, 68R05. Key words and phrases. Difference set, prime power conjecture, An lattice, perfect code, tiling, group splitting, dominating set, covering code. This work was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia (grants TR32040 and III44003), and by the EU COST action IC1104. 1For a nice account of the theory and an extensive list of references see [4], and also [17, 18, 19]. For some of the applications of difference sets in communications and coding theory see, e.g., [1, 23, 31, 7, 10]. 1
ˇ ´ MLADEN KOVACEVI C
2
is a variation of the ℓ1 distance n
1X 1 |xi − yi |, d(x, y) = kx − yk1 = 2 2
(1.2)
i=0
where x = (x0 , x1 , . . . , xn ), y = (y0 , y1 , . . . , yn ). The constant 1/2 is taken for convenience because kx − yk1 is always even for x, y ∈ An . Distance d also represents the graph distance in An ; namely, if Γ(An ) is a graph with the vertex set An and with edges joining neighboring points (i.e., points at distance 1 under d), then d(x, y) is the length of the shortest path between x and y in Γ(An ). 2 Ball of radius 1 around x ∈ An contains 2 n+1 2 +1 = n +n+1 points of the form x+fi,j , where fi,j is a permutation of (1, −1, 0, . . . , 0) having a 1 at the i’th coordinate, a −1 at the j’th coordinate, and zeros elsewhere (with the convention fi,i = 0). In two dimensions we can visualize this ball as a hexagon, and in three as a so-called cuboctahedron (Fig. 1). In general, convex interior of the points in the ball forms a highly symmetrical polytope having the following interesting property, among many others – the distance of all the vertices from the center is equal to the distance between any two neighboring vertices.
Figure 1. Balls of radius 1 in (A2 , d) – hexagon, and in (A3 , d) – cuboctahedron. Ball of radius r around x ∈ An contains all the points with integral coordinates in the convex interior of {x + rfi,j }. For studying packing problems, it is usually simpler to visualize Zn instead of an arbitrary lattice. In our case there is a trivial map that makes the transition to Zn and back very easy. For x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ Zn , define the metric X X (1.3) d+ (x, y) = max (xi − yi ), (yi − xi ) . i: xi >yi
i: xi yi
because
(1.5)
Pn
i=0 xi
=
n X
(yi − xi )
i=0 xi yi ′
xi 0, xj1 , . . . , xjn−k ≤ 0, where k ∈ {0, . . . , n} and {i1 , . . . , ik , j1 , . . . , jn−k } = {1, . . . , n}. It is clear that the number of (k) points in one subset depends only on k; denoting it by Mn (r), we can write Mn (r) = P n (k) k Mn (r). Observe next that the “positive mass” is distributed on the coordinates xi1 , . . . , xik independently of the way the “negative mass” is distributed over xj1 , . . . , xjn−k , (k)
and hence Mn (r) is equal to the product of the number of vectors (xi1 , . . . , xik ) ∈ Zk with P xiu > 0 and ku=1 xiu ≤ r, and the number of vectors (xj1 , . . . , xjn−k ) ∈ Zn−k with xjv ≤ 0 Pn−k and v=1 xjv ≥ −r. The first number is precisely kr , and the second r+n−k n−k . 2. Difference sets as sublattices of An
In the following, when using concepts from graph theory in our setting, we have in mind the graph representation Γ(An ) of An , as introduced above. An (r, i, j)-cover (or (r, i, j)covering code) in a graph Γ = (V, E) [2] is a set of its vertices S ⊆ V with the property that every element of S is covered by exactly i balls of radius r centered at elements of S, while every element of V \ S is covered by exactly j such balls. Special cases of such sets, namely (1, i, j)–covers, have also been studied in the context of domination theory in graphs [30]. In coding theory, (r, 1, 1)-covers are known as r-perfect codes. An independent set in a graph Γ = (V, E) is a subset of its vertices I ⊆ V , no two of which are adjacent in Γ. 2.1. The general case. The proof of the following theorem uses the same method that was employed to prove the connection between lattice tilings and group splitting [14, 28] (see also [29]). In fact, planar difference sets (λ = 1) are an instance of splitting sequences. Theorem 2.1. A (v, k, λ)-difference set exists if and only if the lattice Ak−1 contains a (1, 1, λ)-covering sublattice. Proof. Suppose that D = {d0 , d1 , . . . , dk−1 } is a (v, k, λ)-difference set in a group G. Observe the sublattice ) ( k−1 X xi di = 0 (2.1) L = x ∈ Ak−1 : i=0
where xi di denotes the sum (in G) of |xi | copies of di if xi > 0, and of −di if xi < 0. L is a (1, 1, in An . To see this, consider a point y = (y0 , y1 , . . . , yk−1 ) ∈ / L, meaning Pλ)-cover n that i=0 yi di = a ∈ G, a 6= 0. The neighbors of y are of the form y + fi,j , i 6= j. Since D is a difference set, we know that −a ∈ G can be written as a difference of the elements from D in exactly λ different ways, meaning that there are λ different pairs (s, t) for which ds − dt = −a, dsP , dt ∈ D. ForPevery such pair observe the point zs,t = y + fs,t . We have zs,t ∈ L because ni=0 zi di = ni=0 yi di + ds − dt = a − a = 0. Therefore, there are exactly λ points in the lattice L that are adjacent to y, i.e., such that balls of radius 2This fact is mentioned in [11] for the case n = 2, though the interpretation via the metric d is not +
given.
4
ˇ ´ MLADEN KOVACEVI C
1 around them cover y. To show that the elements of L are covered only by the balls around themselves (i.e., that L is an independent set in Γ(An )), observe that if there were two points at distance 1 in L, then by the same argument as above we would obtain that ds − dt = 0, i.e., ds = dt for some s 6= t, which is a contradiction because |D| = k. For the other direction, assume that L′ is a (1, 1, λ)-covering sublattice of Ak−1 . Define the group G = Ak−1 /L′ , and observe D = {d0 , d1 , . . . , dk−1 } ⊆ G, where di = [fi,0 ] ≡ fi,0 + L′ are the cosets (elements of G). Let us first assure that all the di ’s are distinct. Suppose that ds = dt for some s 6= t. This implies that ds − dt = [fs,t ] = [0], which means that fs,t ∈ L′ . But since 0 ∈ L′ , and 0 and fs,t are at distance 1, this would contradict the fact that L′ is independent. Hence, |D| = k. Now take any nonzero element of G, say [y], y ∈ / L′ . By assumption, y is covered by exactly λ elements of L′ , i.e., y + fs,t ∈ L′ for exactly λ vectors fs,t. Since fs,t = fs,0 − ft,0 , this is equivalent to saying that dt − ds = [ft,0 ] − [fs,0 ] = [y]. Therefore, every nonzero element of G can be written as a difference of two elements from D in exactly λ different ways. Geometrically, the theorem states that the balls of radius 1 around the points of the sublattice L overlap in such a way that every point that does not belong to L is covered by exactly λ balls. (The elements of L, i.e., centers of the balls, are covered by one ball only, and hence this notion is different than that of multitiling [13].) Example 1. D = {0, 1, 2} is a (4, 3, 2)-difference set in the cyclic group Z4 (integers modulo 4). A (1, 1, 2)-covering sublattice L ⊂ A2 corresponding to this difference set is illustrated in Fig. 2. Points in L are depicted as black, and those in A2 \ L as white dots. For illustration, Fig. 3 shows an example of a (1, 2, 2)-covering sublattice (which does not correspond to any difference set). △
Figure 2. A (1, 1, 2)-covering sublattice of A2 . Remark 2.2. We can interpret the lattice L as an error-correcting/detecting code, in which case the points in L are called codewords. When λ = 1, the code can correct a single error because the balls of radius one around codewords do not overlap and the minimum distance of the code is three (here by a single error we mean the addition of a vector fi,j for some i, j, i 6= j, to the “transmitted” codeword x ∈ L). For λ > 1, however, it can only detect a single error reliably. Note also that increasing λ increases the density of the code/lattice L in Ak−1 , but does not affect its error-detection capability. The densest such lattice is therefore obtained for λ = k (that this is the maximum value follows from λ(v − 1) = k(k − 1) and k ≤ v); it corresponds to the trivial (v, v, v)-difference set D = G in an arbitrary group G. △
DIFFERENCE SETS AND CODES IN An LATTICES
5
Figure 3. A (1, 2, 2)-covering sublattice of A2 . Note that we have not specified the order of the elements of the difference set D when defining the corresponding lattice L in (2.1). Therefore, more than one lattice can be defined via D, but they are all identical up to a permutation of the coordinates. Note also that if we write d′i = zdi + g instead of di in (2.1), where z is a fixed integer coprime with v and g is a fixed element of G, identical lattice is obtained because k−1 X
(2.2)
i=0
xi di = 0
⇔
k−1 X
xi d′i = 0
i=0
Pk−1
which follows from i=0 xi = 0 and gcd(z, v) = 1. Let us recall some terminology. Two difference sets D and D ′ in an Abelian group G are said to be equivalent [4, Rem. 1.11, p. 302] if D ′ = {zd + g : d ∈ D}, for some z ∈ Z coprime with v and some g ∈ G. Two codes C and C ′ of length m over an alphabet A are equivalent [24, p. 40] if there exist m permutations of A, π1 , . . . , πm , and a permutation σ over {1, . . . , m} such that (2.3) C ′ = σ(π1 (x1 ), . . . , πm (xm )) : (x1 , . . . , xm ) ∈ C . We then have the following:
Proposition 2.3. If two difference sets D and D ′ are equivalent, then the corresponding codes (defined as in (2.1)) are equivalent. In fact, the πi ’s are necessarily identity maps, only σ is relevant here. 2.2. Planar difference sets as perfect codes. An interesting special case of Theorem 2.1 is obtained by considering planar difference sets (λ = 1). Recall that an r-perfect code in a discrete metric space (U, d) is a subset C ⊆ U with the property that balls of radius r around the “codewords” from C are disjoint and cover the entire space U . In the above terminology r-perfect codes are (r, 1, 1)-covers; 1-perfect codes are also called efficient dominating sets in graph theory [3]. When U is a vector space and C its subspace, C is said to be a linear code. The same terminology is often used for lattices, namely, C is called a linear code if it is a sublattice of the lattice in question. Corollary 2.4. Planar difference set of order n exists if and only if the space (An , d) admits a linear 1-perfect code (i.e., an efficient dominating sublattice).
6
ˇ ´ MLADEN KOVACEVI C
Existence of such codes when n is a prime power follows from the existence of the corresponding difference sets [26], but the necessity of this condition is open and is equivalent to the prime power conjecture. Conjecture 2.5 (Prime power conjecture). There exists a linear 1-perfect code in (An , d) (or, equivalently, in (Zn , d+ )) if and only if n is a prime power. △ A stronger conjecture would claim the above even for nonlinear codes. Example 2. Consider the (13, 4, 1)-difference set D = {0, 1, 3, 9} ⊂ Z13 . The corresponding 1-perfect code in (A3 , d) is illustrated in Fig. 4. The figure shows the intersection of A3 with the plane x0 = 0; the intersections of a ball of radius 1 in A3 with the planes x0 = const are shown in Fig. 5 for clarification. △
Figure 4. 1-perfect code in (A3 , d).
Figure 5. Intersections of a ball in (A3 , d) with the planes x0 = const. Another important unsolved conjecture in the field is the following: All Abelian planar difference sets live in cyclic groups [4, Conj. 7.7, p. 346]. Since the group G containing the difference set which defines the code L is isomorphic to An /L (see the proof of Theorem 2.1), the statement that G is cyclic, i.e., that it has a generator, is equivalent to the following: Conjecture 2.6 (All Abelian planar difference sets are cyclic). Let L be a 1-perfect code in (An , d). Then the period of L in An along the direction fi,j is equal to n2 + n + 1 for at least one vector fi,j , (i, j) ∈ {0, 1, . . . , n}2 . △
DIFFERENCE SETS AND CODES IN An LATTICES
7
The cyclic case. In the rest of this subsection we restrict our attention to cyclic planar difference sets of order n, i.e., it is assumed that the group we are working with is Zv , v = n2 + n + 1; as mentioned above, this in fact might not be a restriction at all. So let D = {d0 , d1 , . . . , dn } ⊂ Zv be a difference set and L the corresponding code (see (2.1)). We will assume that d0 = 0, d1 = 1. (This is not a loss in generality because if D is a difference set, there exist two elements, say d0 , d1 ∈ D, such that d1 − d0 = 1. Then we can take the equivalent difference set D ′ = {di − d0 : di ∈ D} which obviously contains 0 and 1.) In this case the generator matrix of the code (lattice) L has the following form v 0 0 ··· 0 −d2 1 0 · · · 0 (2.4) B(L) = −d3 0 1 · · · 0 , .. .. .. .. . . . . . . . −dn 0 0 · · · 1
i.e., the codewords are the vectors x = ξ · B(L), ξ ∈ Zn (the The generator matrix of the dual lattice L∗ is 1 d d3 2 ··· v v v 0 1 0 · · · ∗ −1 (2.5) B(L ) = B(L) = 0 0 1 · · · .. .. .. . . . . . . 0 0 0 ···
vectors are written as rows). dn v
0 0 . .. . 1
We have disregarded above the 0-coordinate because d0 = 0. Therefore, B(L) is in fact a generator matrix of the corresponding code in (Zn , d+ ) (see Lemma 1.1).
Finite alphabet. By taking the codewords of L modulo v = n2 + n + 1, one obtains a finite code in Znv defined by the generator matrix (over Zv ) −d2 1 0 · · · 0 −d3 0 1 · · · 0 (2.6) .. .. .. . . .. . . . . . . −dn 0 0 · · ·
1
This code is of length n, has v n−1 codewords, and is 1-perfect (under the obvious “modulo v version” of the d+ metric). It is also systematic, i.e., the information sequence itself is a part of the codeword. The “parity check” matrix of the code is H = 1 d2 · · · dn . Thus, the codewords are all those vectors x = (x1 , . . . , xn ) ∈ Znv for which H ·xT = 0 mod v, and the syndromes T = d −d . of the correctable error vectors fi,j are H · fi,j i j 3. r-perfect codes in (An , d) A natural question related to the results of Section 2.2 is whether there exist r-perfect codes in (An , d) for r > 1. For example, such codes exist for any r in dimensions one and two: it is easy to verify that the sublattice of A1 spanned by the vector (−2r − 1, 2r + 1) is an r-perfect code in A1 , and the sublattice of A2 spanned by the vectors (−2r − 1, r, r + 1), (−r − 1, 2r + 1, r) is an r-perfect code in A2 (for a study of the two-dimensional case see also [9]). In higher dimensions, however, it does not seem to be possible to tile (An , d) with balls of radius r > 1. We shall not be able to prove this claim here, but Theorem 3.2 below is a step in this direction. Denote by Bn (x, r) = {y ∈ Zn : d+ (x, y) ≤ r} the ball of radius r around x in (Zn , d+ ). Let Dn (r) be the body in Rn defined as the union of unit cubes translated to the points
8
ˇ ´ MLADEN KOVACEVI C
Figure 6. Bodies in R2 corresponding to a ball of radius 2 in Z2 , d+ : The cubical tile (left) and the convex interior (right). S of Bn (0, r), namely, Dn (r) = y∈Bn (0,r) (y + [−1/2, 1/2]n ) (see Fig. 6). Let also Cn (r) be the body defined as the convex interior (in Rn ) of the points in Bn (0, r). Lemma 3.1. The volumes of the bodies Dn (r) and Cn (r) are given by (3.1) (3.2)
Vol(Dn (r)) = Vol(Cn (r)) =
min{n,r}
X
k=0 n r 2n
n!
n
n k
r r+n−k k n−k
.
Furthermore, limr→∞ Vol(Cn (r))/ Vol(Dn (r)) = 1. Proof. Since Dn (r) consists of unit cubes, its volume is Vol(Dn (r)) = |Bn (x, r)|, which gives the above expression by Lemma 1.2. The volume of Cn (r) can be computed by the method similar to the proof of Lemma 1.2, namely, we observe the intersection of Cn (r) with the orthant x1 , . . . , xk > 0, xk+1 , . . . , xn ≤ 0, where k ∈ {0, of this intersection is the product of the volumes of . . . , n}. The volume P the k-simplex (x , . . . , x ) : x > 0, x ≤ r , which is known to be r k /k!, and of the i i k 1 P (n − k)-simplex (xk+1 , . . . , xn ) : xi ≤ 0, xi ≥ −r , which is r n−k /(n − k)!. Therefore, k rn−k 2 P P Vol(Cn (r)) = nk=0 nk rk! (n−k)! , which, together with the identity nk=0 nk = 2n n , gives (3.2). The asymptotic behavior (when r → ∞) of Vol(Dn (r)) is easily found to be Vol(Dn (r)) ∼ r n 2n by using the fact that kr ∼ r k /k!. n! n Theorem 3.2. There are no r-perfect codes in (An , d), n ≥ 3, for large enough r, i.e., for r ≥ r0 (n).
Proof. The proof is based on the same idea as the one for r-perfect codes in Zn under ℓ1 (also termed Manhattan or taxi) distance [12]. First observe that an r-perfect code in (Zn , d+ ) would induce a tiling of Rn with Dn (r), and a packing with Cn (r). The relative efficiency of the latter with respect to the former is defined as the ratio of the volumes of these bodies, Vol(Cn (r))/ Vol(Dn (r)), which by Lemma 3.1 tends to 1 as r grows indefinitely. This has the following consequence: If an r-perfect code exists in (Zn , d+ ) for arbitrarily large r, then there exists a tiling of Rn by translates of the body Dn (r) for arbitrarily large r, which further implies that a packing of Rn by translates of the body Cn (r) exists which has efficiency arbitrarily close to 1. But then there would also be a packing by Cn (r) of efficiency 1, i.e., a tiling (in [12, Appendix] it is shown that there exists a packing whose density is the supremum of the densities of all possible packings with a given body). This is a contradiction. Namely, by [25, Thm 1] a necessary condition
DIFFERENCE SETS AND CODES IN An LATTICES
9
for a convex body to be able to tile space is that it be a centrally symmetric polytope3 with centrally symmetric facets, which the polytope Cn (r) fails to satisfy for n ≥ 3. For example,the facet which is thePintersection of Cn (r) with the hyperplane x1 = −r is the simplex (x2 , . . . , xn ) : xi ≥ 0, ni=2 xi ≤ r , a non-centrally-symmetric body. In summary, we have shown that linear r-perfect codes in (An , d) exist for: • n = 1, r arbitrary, • n = 2, r arbitrary, • n ≥ 3 a prime power, r = 1. The statement that these are the only cases (apart from the trivial one r = 0), even if nonlinear codes are allowed, is a further strengthening of the prime power conjecture. It should also be contrasted with the Golomb-Welch conjecture [12] (see also, e.g., [15, 16]) stating that r-perfect codes in Zn under ℓ1 distance exist only in the following cases: 1) n ∈ {1, 2}, r arbitrary, and 2) r = 1, n arbitrary. 4. Applications in coding theory 4.1. Permutation channels. A permutation channel [21, 22] over an alphabet A is a communication channel that takes sequences of symbols from A as inputs, and for any input sequence outputs a random permutation of this sequence. This channel is intended to model packet networks based on routing in which the receiver cannot rely on the packets being delivered in any particular order, as well as several other communication scenarios where a similar effect occurs, such as systems for distributed storage, data gathering in wireless sensor networks, etc. It was shown in [22] that the appropriate space in which error-correcting codes for such channels should be defined is (∆nℓ , d), where ) ( n X n+1 n xi = ℓ (4.1) ∆ℓ = (x0 , x1 , . . . , xn ) ∈ Z≥0 : i=0
is the discrete standard simplex, and d is the metric given by (1.2). Notice that ∆nℓ is just the translated An lattice restricted to the nonnegative orthant. This restriction is the reason that this space lacks some nice properties that are usually exploited when studying bounds on codes, packing problems, and the like. In order to study the underlying geometric problem, one can disregard these restrictions and investigate the corresponding problems in (An , d). The same approach is employed for some other types of codes; for example, studying the geometry of codes for flash memories reduces to packing problems in Zn , see, e.g., [5, 6, 27]. 4.2. Particle insertion/deletion channels. Consider the following channel model. The transmitter sends xi particles (or packets) in the i’th time slot. The particles are assumed identical, implying that the transmitted sequence can be identified with a sequence of nonnegative integers (x1 , . . . , xn ) ∈ Zn≥0 . In the channel, some of the particles can be lost (deletions), while new particles can appear (insertions) from the surrounding medium. Additionally, one can also allow delays of the particles in the model [20]. (Notice that if a particle is delayed, this can be thought of as it being deleted, while another particle is being inserted several time slots later.) Such channels are of interest in molecular communications, discrete-time queuing systems, etc. We want to guarantee that all patterns of ≤ r insertions and ≤ r deletions can be corrected at the receiving side. In other words, we want to be able to recover from all 3A polytope P ⊂ Rn is centrally symmetric if its translation P˜ = P − x satisfies P˜ = −P˜ for some
x ∈ Rn .
10
ˇ ´ MLADEN KOVACEVI C
patterns of errors such that4 max{#deletions, #insertions} ≤ r. It is not difficult to see that this will be achieved if and only if the d+ -balls of radius r around codewords are disjoint. In other words, the decoding regions are in this case defined precisely by balls in d+ metric. Hence, packing/tiling problems in (Zn , d+ ) are indeed relevant for studying and designing good codes for channels of this type. 4.3. Channels with ordered alphabets. Observe the channel with input and output alphabet A = {0, 1, . . . , q − 1} and think of the symbols in A as voltage levels (e.g., in baseband digital signal transmission, or in multi-level flash memory [6, 27]). Assume that we wish to assure that the receiver will be able to recover the signal whenever the total voltage drop (across all symbols of a particular codeword) and the total voltage increase caused by noise are at most r. This situation is essentially identical to the one in the previous example, and it is clear that the codes in (Zn , d+ ) (more precisely their restrictions to An ) are adequate constructions for error-correction in this scenario. Acknowledgment The author would like to thank Dr Moshe Schwartz for reading the manuscript and providing several useful comments. References 1. M. D. Atkinson, N. Santoro, and J. Urrutia, “Integer Sets with Distinct Sums and Differences and Carrier Frequency Assignments for Nonlinear Repeaters,” IEEE Trans. Commun., vol. 34, no. 6, pp. 614–617, Jun. 1986. 2. M. A. Axenovich, “On Multiple Coverings of the Infinite Rectangular Grid with Balls of Constant Radius,” Discrete Math., vol. 268, no. 1–3, pp. 31–48, Jul. 2003. 3. D. W. Bange, A. E. Barkauskas, and P. J. Slater, “Efficient Dominating Sets in Graphs,” in: R. D. Ringeisen and F. S. Roberts (eds.), Applications of Discrete Mathematics, SIAM, Philadelphia, 1988, pp. 189–199. 4. T. Beth, D. Jungnickel, and H. Lenz, Design Theory, 2nd ed., Cambridge University Press, 1999. 5. S. Buzaglo and T. Etzion, “Tilings with n-Dimensional Chairs and Their Applications to Asymmetric Codes,” IEEE Trans. Inform. Theory, vol. 59, no. 3, pp. 1573–1582, Mar. 2013. 6. Y. Cassuto, M. Schwartz, V. Bohossian, and J. Bruck, “Codes for Asymmetric Limited-Magnitude Errors with Application to Multilevel Flash Memories,” IEEE Trans. Inform. Theory, vol. 56, no. 4, pp. 1582–1595, Apr. 2010. 7. C. Chen, B. Bai, and X. Wang, “Construction of Nonbinary Quasi-Cyclic LDPC Cycle Codes Based on Singer Perfect Difference Set,” IEEE Commun. Lett., vol. 14, no. 2, pp. 181–183, Feb. 2010. 8. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Springer, 1999. 9. S. I. R. Costa, M. Muniz, E. Agustini, and R. Palazzo, “Graphs, Tessellations, and Perfect Codes on Flat Tori,” IEEE Trans. Inform. Theory, vol. 50, no. 10, pp. 2363–2377, Oct. 2004. 10. M. Esmaeili and M. Javedankherad, “4-Cycle Free LDPC Codes Based on Difference Sets,” IEEE Trans. Commun., vol. 60, no. 12, pp. 3579–3586, Dec. 2012. 11. T. Etzion, “Sequence Folding, Lattice Tiling, and Multidimensional Coding” IEEE Trans. Inform. Theory, vol. 57, no. 7, pp. 4383–4400, Jul. 2011. 12. S. W. Golomb and L. R. Welch, “Perfect Codes in the Lee Metric and the Packing of Polyominoes,” SIAM J. Appl. Math., vol. 18, no. 2, pp. 302–317, Mar. 1970. 13. N. Gravin, S. Robins, and D. Shiryaev, “Translational Tilings by a Polytope, with Multiplicity,” Combinatorica, vol. 32, no. 6, pp. 629–648, Dec. 2012. 14. W. Hamaker, “Factoring Groups and Tiling Space,” Aequationes Math., vol. 9, no. 2-3, pp. 145–149, 1973. 15. P. Horak, “Tilings in Lee Metric,” European J. Combin., vol. 30, no. 2, pp. 480–489, Feb. 2009. 16. P. Horak, “On Perfect Lee Codes,” Discrete Math., vol. 309, no. 18, pp. 5551–5561, Sep. 2009. 17. D. Jungnickel, “Difference Sets,” in: Contemporary Design Theory: A Collection of Surveys (Eds. J. H. Dinitz and D. R. Stinson), Wiley, New York, 1992, pp. 241–324. 4Notice that this approach is slightly different from the standard one where one requires that the
patterns of errors with (#deletions + #insertions) ≤ r are correctable. Thus, we treat deletions and insertions separately here because these impairments are of different nature and different origin.
DIFFERENCE SETS AND CODES IN An LATTICES
11
18. D. Jungnickel and B. Schmidt, “Difference Sets: An Update,” in: Geometry, Combinatorial Designs and Related Structures (Eds. J. W. P. Hirschfeld, S. S. Magliveras, and M. J. de Resmini), pp. 89–112, Cambridge University Press, 1997. 19. D. Jungnickel and B. Schmidt, “Difference Sets: A Second Update,” Rend. Circ. Mat. Palermo Serie 2 Suppl., vol. 53, pp. 89–118, 1998. 20. M. Kovaˇcevi´c and P. Popovski, “Zero-Error Capacity of a Class of Timing Channels,” IEEE Trans. Inform. Theory, vol. 60, no. 11, pp. 6796–6800, Nov. 2014. 21. M. Kovaˇcevi´c and D. Vukobratovi´c, “Multiset Codes for Permutation Channels,” preprint available at arXiv:1301.7564. 22. M. Kovaˇcevi´c and D. Vukobratovi´c, “Perfect Codes in the Discrete Simplex,” Des. Codes Cryptogr., to appear, DOI: 10.1007/s10623-013-9893-5. 23. A. W. Lam and D. V. Sarwate, “On Optimum Time-Hopping Patterns,” IEEE Trans. Commun., vol. 36, no. 3, pp, 380–382, Mar. 1988. 24. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company, 1977. 25. P. McMullen, “Convex Bodies Which Tile Space by Translation,” Mathematika, vol. 27, no. 1, pp. 113–121, Jun. 1980. 26. J. Singer, “A Theorem in Finite Projective Geometry and Some Applications to Number Theory,” Trans. Amer. Math. Soc., vol. 43, pp. 377–385, 1938. 27. M. Schwartz, “Quasi-Cross Lattice Tilings with Applications to Flash Memory,” IEEE Trans. Inform. Theory, vol. 58, no. 4, pp. 2397–2405, Apr. 2012. 28. S. Stein, “Packings of Rn by Certain Error Spheres,” IEEE Trans. Inform. Theory, vol. 30, no. 2, pp. 356–363, Mar. 1984. 29. S. Stein and S. Szab´ o, Algebra and Tiling: Homomorphisms in the Service of Geometry, The Mathematical Association of America, 1994. 30. J. A. Telle, “Complexity of Domination-Type Problems in Graphs,” Nordic J. Comput., vol. 1, pp. 157–171, 1994. 31. E. J. Weldon, Jr., “Difference-Set Cyclic Codes,” Bell Syst. Tech. J., vol. 45, pp. 1045–1055, 1966. ´a Department of Electrical Engineering, University of Novi Sad, Trg Dositeja Obradovic 6, 21000 Novi Sad, Serbia. E-mail address: [email protected]
