Flat Tori, Lattices and Spherical Codes - ITA @ UCSD
Flat Tori, Lattices and Spherical Codes Sueli I. R. Costa
Cristiano Torezzan
Antonio Campello
Vinay A. Vaishampayan
Institute of Mathematics School of Applied Sciences Institute of Mathematics AT&T Shannon Laboratory University of Campinas, Brazil University of Campinas, Brazil University of Campinas, Brazil Florham Park, NJ, USA [email protected] [email protected] [email protected] [email protected]
Abstract—The foliation of a sphere in an even number of dimensions by flat tori can be used to construct discrete spherical codes and also homogeneous curves for transmitting a continuous alphabet source over an AWGN channel. In both cases the performance of the code is related to the packing density of specific lattices and their orthogonal sublattices. In the continuous case the packing density of curves relies also on the search for projection lattices with good packing density. We present here a survey of this topic including some recent research results and perspectives. Keywords—Spherical codes, group codes, flat torus, lattices, Gaussian channel, codes on graphs, continuous alphabet source.
I.
I NTRODUCTION
Group codes as introduced by Slepian [1] and developed in subsequent articles [2], [3], [4], [5] are defined as finite sets on an n-dimensional sphere generated by the action of a group of orthogonal matrices. Geometrically uniform codes introduced by Forney [6] generalize this concept by considering also infinite sets of points in Euclidean space having a transitive symmetry group. We may say that the main course of what is presented here follows this concept in the context of metric spaces [7]: a signal set S ⊂ X is a geometrically uniform code if and only if for s, t in S there is an isometry f (depending on s, t) in X such that f (s) = t and f (S) = S. Geometrically uniform codes capture the highly desirable properties that come from homogeneity: a distance profile and error probability that is identical for every codeword, and Voronoi regions that are congruent to each other. One recurrent support metric space considered here is the n dimensional flat torus, obtained by identifying the opposite sides of a n dimensional box, more precisely defined as a quotient T = Rn /GT where GT is the group of translations generated by n independent vectors. A 2 − D flat torus can be visualized as a standard torus in the 3-dimensional space, but it can be distinguished from the latter by being locally flat i.e. like a piece of a plane. It can only be realized isometrically as a 2-dimensional surface in R4 which is contained in a 3-dimensional sphere. In this paper we present an overview of the strong connection between flat tori, lattices and group codes on a sphere, summarize some previous works and also point out recent results and perspectives of applications. Section II describes the foliation of a sphere in even dimensions, n = 2L, by L dimensional flat tori. Inequalities relating the distances on a flat torus in R2L and on its associated Work partially supported by FAPESP under grants 2009/18337-6, 2007/56052-8, 2012/09167-2 and CNPq 309561/2009-4
hyperbox in RL are also presented for later use. Section III is a strongly geometrical approach to commutative group codes presenting their connections with flat tori and quotients of lattices that allows the establishment of specific upper bounds for those codes. Some results on constructions which may approach those bounds and for optimal commutative group codes are discussed. Some remarks on commutative group codes considered on graphs are also included. Section IV-A summarizes a construction of spherical codes on layers of flat tori with some comparisons with well known spherical codes. In Section IV-B the homogeneous structure of flat tori and lattices come together again, now as a proposal for transmitting a continuous alphabet source over an AWGN channel. The search for projection lattices with good packing density is of significance here. Finally in Section V some concluding remarks including recent research and perspectives are drawn. II.
F LAT T ORI
The unit sphere S 2L−1 ⊂ R2L can be foliated by flat tori (Clifford Tori) as follows. For each unit vector c = (c1 , c2 , .., cL ) ∈ L X L−1 c2i = 1, and u = (u1 , u2 , . . . , uL ) ∈ RL , let S , ci > 0, i=1
Φc : RL → R2L be defined as u) = Φc (u
u1 u1 uL uL c1 cos( ), c1 sin( ), . . . , cL cos( ), cL sin( ) . c1 c1 cL cL (1)
The image of this periodic mapping Φc is the torus Tc , a flat L-dimensional surface on the unit sphere S 2L−1 . Tc is also the image of an L-dimensional box Pc , u ∈ RL ; 0 ≤ ui < 2πci }, 1 ≤ i ≤ L, Pc = {u
(2)
in which Pc is injective.
For c ∈ S L−1 and ci ≥ 0, if ci = 0 for some 1 ≤ i ≤ L, we may replace in (1) both coordinates related to ci by 0 and obtain a degenerated flat torus Tc , which is an embedding of a (L − k)-dimensional box in R2L , where k is the number of zero coordinates of c . The Gaussian curvature of a torus Tc is zero and Tc can be cut and flattened into the box, Pc , just as a cylinder in R3 can be cut and flattened into a 2-dimensional rectangle [8]. Since the inner product h∂Φc /∂ui , ∂Φc /∂uj i = δij , where δij is the Kronecker delta function, the application Φc is a local isometry, which means that any measure of length, area and volume up to dimension L − k on Tc is the same of the corresponding pre-image in the box Pc .
by u ), Φc (vv )|| = 2 ||Φc (u
S3
r X
c2i sin2 (
ui − vi ) 2ci
(4)
and it is bounded according to the next proposition [10]. Proposition 2. Let c = (c1 , c2 , .., cL ) ∈ S 2L−1 , ci > 0, cξ = u − v k for u , v ∈ Pc . Suppose 0 < ∆ ≤ min ci 6= 0, ∆ = ku
1≤i≤L
cξ /2, then
2∆ ≤ sin π
III.
∆ 2cξ
u) − Φc (vv )k ≤ 2cξ ≤ kΦc (u
sin ∆ 2 ≤∆ 2
C OMMUTATIVE G ROUP C ODES , F LAT T ORI AND L ATTICES
A. Commutative Group Codes
Pc
Let On be the multiplicative group of orthogonal n × n matrices and let Gn (M ) be the set of all order M commutative subgroups in On .
2πc2
A spherical commutative group code C is a set of M vectors which is the orbit of an initial vector u on the unit sphere S n−1 ⊂ Rn under the action of subgroup G of Gn (M ), u = {gu u, g ∈ G} . i.e. C := Gu
u
2πc1
The minimum distance in C is: d :=
c = (c1, c2)
min
x, y ∈ C x 6= y
||x − y|| =
min
gi 6= I ∈ G
||gi x − x||,
where ||.|| denotes the standard Euclidean norm. A canonical form for the a commutative group G ∈ Gn (M ) can be obtained from the following result.
S
Proposition 3. [[11],p 292] A commutative group G = {Oi }M i=1 of n × n orthogonal real matrices can be carried by one and the same real orthogonal transformation Q into the diagonal block canonical form QT Oi Q =
1
2πbi1 2πbiq ), . . . , R( ), µ2q+1 (i), . . . , µn (i)], (5) M M where bij are integers, the blocks R(a) are the 2-dimensional rotations cos(a) − sin(a) , R(a) = sin(a) cos(a) [R(
Fig. 1. Illustration of a torus layer Tc , c = (0.6, 0.8).
We say that the family of flat tori Tc and their degenerations, with c = (c1 , c2 , .., cL ), kcck = 1, ci ≥ 0, defined above is a foliation on the unit sphere of S 2L−1 ⊂ R2L . This means that any vector of S2N −1 belongs to one and only one of these flat tori. The following results [9] allow to relate the distances between two points in RL and their spherical image on flat tori in R2L Proposition 1. Let Tb and Tc be two flat tori, defined by unit vectors b and c with non negative coordinates. The minimum distance d(Tc , Tb ) between two points on these flat tori is !1/2 L X . (3) (ci − bi )2 d(Tc , Tb ) = kcc − b k =
and µl (i) = ±1 with l = 2q + 1, . . . , n. The next proposition [12] describes the geometric locus of a commutative group code. For even dimension this locus is always contained in a flat torus. Proposition 4. Every commutative group code of order M is, up to isometry, equal to a spherical code X whose initial vector is u = (u1 , . . . , un ) and its points have the form (R(ai1 )(u1 , u2 ), . . . , R(aiq )(u2q−1 , u2q ), µ2q+1 (i)u2q+1 , . . . , µn (i)un ),
1)
i=1
u ) and Φc (vv ) on the The distance between two points Φc (u same torus Tc , defined by a vector c = (c1 , · · · , cL ), is given
2πbij . Moreover, M If n = 2L, X is contained in the flat torus Tc , c = (c1 , . . . , cL ) where satisfies c 2i = u22i−1 + u22i . If n = 2L + 1 and X is substantial, X = X1 ∪ X2 , where Xi is contained in the plane Pi = {(x1 , . . . , x2L+1 ) ∈ R2L+1 ; x2L+1 = (−1)i un }.
where aij =
2)
Also, Xi is contained in the torus Tc of a sphere in R2m with radius (1−u2n )1/2 , where c 2i = u22i−1 +u22i . B. Lattice connections We say that a 2L-dimensional commutative group code is free from reflection blocks if its generator matrix group, considered as Proposition 3, satisfies 2L = 2q = n. By reflection blocks, we refer to the 2-dimensional blocks −1 0 , ± 0 1 which appear in the canonical form when 2q < n. Commutative group codes in even dimension, whose generator matrices are free from reflections blocks, are directly related to lattices. u we may For such commutative group codes C = Gu consider without loss of generality the initial vector as u = (c1 , 0, c2 , 0, . . . , cL , 0) where c = (c1 , c2 , .., cL ) is a unit vector. We also will consider here ci > 0 that is, codes that are not contained in a hyperplane of R2L . For the rotation angles aij = (2πbij )/M , where 1 ≤ i ≤ M , 1 ≤ j ≤ L as in proposition 4, let v i = (ai1 , . . . aiL ), 1 ≤ i ≤ M and the lattice Λ defined as the set of all integer combinations Q of v i . Note that Λ contains the orthogonal lattice Λc = (2πci )Z as a sublattice. The connection between these two lattices and u is given next [12]. the group code C = Gu
u with u = (c1 , 0, c2 , 0, . . . , cL , 0), Proposition 5. Let C = Gu c = (c1 , c2 , .., cL ), ||c|| = 1,ci > 0 be a commutative group code in R2L , free from reflection blocks. The inverse image Φc−1 by the torus mapping (1) is the lattice Λ defined as Λ above. Moreover the quotient of lattices is isomorphic to Λc the generator group G.
The last propositions combined with Proposition 2 allow to set bounds for commutative group codes via bounds for spherical packings on flat tori [12]. u of Proposition 7. Every commutative group code C = Gu order M in R2L free from 2 × 2 reflection blocks with initial vector u = (u1 , . . . , u2L ) and minimum distance d satisfies !L 2 1/2 2 ∆Guu π L ΠL π i=1 (u2i−1 + u2i ) M≤ ≤ ∆L , (arcsin d4 )L (arcsin d4 ).L1/2 where ∆Guu is the center density of the lattice Λ associated to the code and ∆L is the maximum center density of a lattice packing in RL . For general commutative group in R2L the lattice packing density can be replaced by the best periodical packing density in RL . Since any packing density in RL can be approached by periodical packing densities as remarked in [14], we can also replace ∆L in the last preposition for DL , by the best center packing density in RL [12]. u of Proposition 8. Every commutative group code C = Gu order M in R2L with initial vector u = (u1 , . . . , u2L ) and minimum distance d satisfies !L 2 2 1/2 π L ΠL DL π i=1 (u2i−1 + u2i ) M≤ ≤ DL , (arcsin d4 )L (arcsin d4 ).L1/2 where DL is the maximum center density of a spherical packing in RL . Bounds for commutative group codes in odd dimensions, n = 2L + 1, can also be obtained [12] by observing that those codes must lie on two parallel hyperplanes and is formed by two equivalent copies of commutative group codes in R2L .
7 14 21 3 10
7
2
17
12
22
17 24 6 13 20
9
4
19
14
24
2 9 16 23 5
6
1
16
11
21
12 19 1 8 15
3
8
13
18
23
22 4 11 18 0
0
5
10
15
20
Fig. 2. Pre-images Φc−1 of two cyclic group codes C = Gu of order M = 25 2π 2π7 in R√4 . On the the initial vector is u = √ left, G = h[R( 25 ), R( 25 ]i and 2π (1/ 2, 0, 1/ √ 2, 0). On the √ right side, G = h[R( 25 ), R( 2π10 ]i and initial 25 vector is u = ( 0.54915, 0, 0.45085, 0), which gives the best commutative group code of this order in R4 [13].
The inverse image through the torus mapping Φc of a commutative group code of order M generated by matrices which may contain 2 × 2 reflection blocks ( 2q < n in Proposition 4) is not always a quotient of lattices. However, from the L-periodicity of Φc in RL , we can write: Proposition 6. The inverse image through the torus mapping u of order M in R2L , Φc of any commutative group code Gu u = (u1 , . . . , u2L ) is a periodic distribution of M points in q L 2 2 the hyperbox Pc ⊂ R , ci = u2i−1 + u2i spanned by the lattice associated to this box.
The torus bounds given in propositions 7 and 8 are tight in the following sense: consider, for instance, the dual inequality of Proposition 8, ! L Y ci DL /M . d ≤ 2 sin i=1
For big M the distance d must be small (from Proposition 2) and the inverse image of the spherical cap of radius d in R2L centered in a point of Tc will be arbitrarily close to the ball of same radius in RL Fig. 3. This means that the best packing in the flat torus will be approached by the best packing in its pre-image in the box Pc and then the upper bounds of the above propositions will be approached. C. Approaching the bound: optimum commutative group codes For small distances δ or big M the search for good commutative group codes may rely on the search of orthogonal ˜ of a lattice Λ with good packing density. For sublattices Λ ˜ let b1 ≤ b2 ≤ . . . bL be length of the each such sublattice Λ P L 2 orthogonal basis vectors, b = b i=1 i and c = (c1 , . . . , cL ) ˜ The where ci = bi /b and the re-scaled lattices 1/bΛ and 1/bΛ. 1/bΛ commutative group code C associated to the quotient ˜ 1/bΛ on the flat torus Tc is a possible choice for a good code,
lattice ΛG (cc) where generator matrix T satisfies the following conditions: 1) 2) 3) 4)
Fig. 3. Inverse images of 4-dimensional balls of several radii though Φc , √ √ c = (2/ 5, 1/ 5). For a small radius it approaches a ball of the same radius in R2 .
particularly if Λ has the best packing density in L-dimensions. See [12] for examples in R6 of how the torus bounds can be approached when M increases. The algebraic group classification and matrix generators for the group code can be found via matrix reduction (Hermite and Smith Normal Forms) [13]. In what follows, C(M, n, d) denotes a code C in Rn with M points and minimum distance equal to d. A C(M, n, d) is said to be optimum if d is the largest minimum distance for a fixed M and n. As it is well known, the minimum distance of a group code C, generated by a finite group G, may vary significantly depending on the choice of the initial vector u. This problem still does not have a general solution, but has been studied in some important special cases, including reflection group codes [15] and permutation group codes [16]. Besides, Biglieri and Elia have shown in [2] that, for a fixed cyclic group code the problem can be formulated as a linear programming problem. They also discussed the efficiency of some of these codes and remarked on the hardness of obtaining the best cyclic group code for a given cardinality M and dimension n. In the search for the best commutative group code C(M, n, d), for fixed values of M and n we must first find the set Gn (M ) of all commutative groups in On of order M and then the best initial vector must be found for each one of those groups. An optimum code will be one which has the largest minimum distance in this set. The total number of Gn (M ) is related with the Euler number of divisors of M and is of order M/2 . It is worth to remark that even isomorphic groups n/2 must be considered, since the resulting minimal distance may vary depending on which representation in On is taken for each group, i.e. two isomorphic groups may generate two non isometric spherical codes, as illustrated in Fig. 2. Our approach to this problem in based on the association between commutative group codes and lattices described here. An important step of the algorithm derived in [13] is to reduce the number of cases to be analysed by discarding isometric codes. This is done via the following proposition. Proposition 9. Every commutative group code C(M, 2L, d), generated by a group G ∈ O2L free of 2 × 2 reflection blocks is isometric to a code obtained as the image by Φc of a
T is in the Hermite Normal Form; det(T ) = M L−1 ; There is a matrix W , with integer elements satisfying W T = M IL , where IL is the L × L identity matrix; The elements of the diagonal of T satisfy T (i, i) = M where ai is a divisor of M and (ai )i · ai (ai+1 · · · aL ) 6 M , ∀i = 1, . . . , L.
As an example of application of the above proposition let us consider M = 128. There are, up to isomorphism, only 4 abstract commutative groups or order 128: {Z128 , Z2 × Z64 , Z4 × Z32 , Z8 × Z16 }. However for n = 2L = {4, 6, 8} there are {2016, 41664, 635376} distinct representations of them in On . After discarding isometric codes by using Proposition 9 we must consider just {71, 2539, 55789} for n = 2L = {4, 6, 8}, respectively [13]. Then the initial vector problem will be solved only for those cases. D. Commutative group codes as codes on graphs Commutative group codes can also be viewed as a graph or a cosset code [17] on a flat torus with the graph distance (minimum number of edges from one vertex to another). They are also geometrically uniform in this context. This is the approach presented in [7]. As an example, consider the codes presented in Figure 2 where each edge of the flat torus box is subdivided into M = 25 segments with the underlined grid associated to this subdivision. Considering also the boundary identification, those grids define a graph on each flat torus 2 with vertices associated to the group Z25 . On the left we have the code generated by the element (b1, b2) = (1, 7), which is a perfect cyclic code in Z25 × Z25 of order M = 25 and minimum distance equal to 7, under the graph (or Lee) distance (a 3-error correcting code). On the right we have the code generated by (b1, b2) = (1, 10), which has minimum distance graph 5. Thus, viewed as graph codes, the code on left on Fig. 2 is better than the code on right in opposition to the performance of their images as spherical codes in R4 . Since both codes are isomorphic, as quotient of lattices, to the cyclic group Z25 they both are associated and present a more geometrical view of a circulant graph. (see Figure 4). This geometrical view may provide tools to analyse circulant and Cayley graphs which are used in parallel computing schemes [18]. IV.
S PHERICAL
CODES IN LAYERS OF TORI AND APPLICATIONS
A. Codes for the Gaussian Channel Although commutative group codes discussed in last section have applications based on their rich structure, those codes are not good in general concerning their the trade-off between distance and number of points, that is, the packing density on the sphere. Flat tori layers can be used to construct spherical codes which combine the good structure of commutative group codes in each layer with a better packing density. A Torus Layer Spherical Code (TLSC) can be generated by a finite set of orthogonal matrices and thus inherited group structure and
0
24
apple-peeling [19], a wrapped [20] and a laminated [21] codes, as illustrated in table I (see also [9] and [10]) for examples of TLSC in dimensions 6, 8 and 48.
1
23
2 3
22
4
21
5
20
19
6
18
7
d 0.5 0.4 0.3 0.2 0.1 0.01
TLSC(4,d) 172 308 798 2,718 22,406 2.27 ×107
TABLE I.
apple-peeling 136 268 676 2,348 19,364 1.97 ×107
wrapped * * * * 17,198 2.31 ×107
laminated * * * * 16,976 2.31 ×107
F OUR - DIMENSIONAL CODE SIZES AT VARIOUS MINIMUM DISTANCES . (*): UNKNOWN VALUES .
8
17 9
16 15
10
B. Curves on torus and continuous alphabet source
11
14 13
12
Fig. 4. The cyclic group code of Figure 2 left viewed as the circulant graph C25 (1, 7).
homogeneity allowing efficient storage and decoding process, which is attached to lattices in the half of the code √dimension. To design theses codes, given a distance d ∈ (0, 2], we first d
c4
d
c3
The homogeneous structure of flat tori and lattices can be used in a proposal for transmitting a continuous alphabet source over an AWGN channel [22], [23]. Consider the problem illustrated in Figure IV-B. A value x from a source with pdf having support [0, 1) is to be transmitted over a Gaussian channel of dimension N (or N orthogonal AWGN channels). The encoder will use a function s : [0, 1) → RN and then transmit the encoded value s (x) over the channel, such that the receiver will observe a noisy vector y = s(x) + z . The objective is to recover an estimate x of the transmitted value, attempting to minimize the mean square error (mse) ˆ 2 ]. If the N -dimensional Gaussian channel has E[(X − X)
d
c2 c1
d
d
d d
2πc12 2πc11
Pc
Fig. 5. Illustration of the construction of a four dimensional torus layer spherical code.
define a collection of tori in S 2L−1 such that the minimum distance between any two of these tori is at least d. This can be done (Proposition 1) by designing a spherical code in RL with minimum distance d and positive coordinates. Then, for each one of these tori, a finite set of points is chosen in RL such that the distance between any two points, when embedded in R2L by the standard parametrization (1), is greater than d, according to Proposition 2. This set of points may belong to a L−dimensional lattice, restricted to a hyperbox Pc (2), chosen to approach a good packing density in RL as described in Section III. The T LSC(2L, d) is the union of the commutative group codes associated to each one of the chosen tori. Figure 5 illustrates the construction of a T LSC(4, d). Those codes are homogeneous in each torus layers and have simplified coding and decoding processes (flat torus decoding in half of the dimension). For not very small distances (or non asymptotic context) a torus layer spherical code may have comparable performance to others well known spherical codes such as an
power P and variance σ 2 , then the average of the transmitted value should be no greater than P i.e. s should satisfy the 2 constraint E[kss(x)k ] ≤ P . This essentially means that the image of [0, 1) by the mapping s needs to be contained withing a sphere of radius P . On the other hand, it can be shown that for low noise, the mse of the scheme represented in Figure IV-B is given by Z 1 −2 2 2 ˆ ˆ 2 ], E[(X − X) ] ≈ σ f (x) ks˙ (x)k dx := Elow [(X − X) 0
(6) The quantity S(x) := ks˙ (x)k is called the stretch of the curve. Summarizing, we want to stretch the function as much as possible, but since s (x) is contained within a sphere, it has to be twisted, or folded. If the distance between the folds of the curve become too small, large errors will occur frequently when the noise is high, and the mse will approach Equation (6) only if the noise is sufficiently low. This is called the threshold effect. Figure 6 displays a picture of small and large errors. For the model described in Figure IV-B, the mse cannot decay faster than O(SN R−N ) [24]. It was shown in [22] that curves on the flat torus determined by the √ vector c = eˆ = (1/ L)(1, . . . , 1) can be used to obtain the correct scaling of the mse with the signal-to-noise ratio in
and consider the bijective mapping fk : Ik → [0, 1) Pk−1 x − j=1 Lenj /Len . fk (x) = Lenk /Len Then the full encoding map s can be defined by s (x) := s Tk (fk (x)), if x ∈ Ik .
Fig. 6. Illustration of small and large errors.
RN , N = 2L. Given a vector a ∈ ZL , the curves considered in [25] are of the type 2π s (x) = φeˆ √ a x(mod 1) , (7) L ax⌉. Due to Proposition (2), it can where a x(mod 1) = a x − ⌊a be shown that the small ball radius of s(x) is approximately the minimum distance of the lattice which is the projection of ZL onto the√subspace a ⊥ , while its stretch is constant and equal ak)2 . Thus, projections of the cubic lattice play to (2π/ L ka an important role on the design of such curves; the denser the projection lattice, the denser the curve on the sphere S 2L−1 . The result in [22] states that if a = (1, a, . . . , aL−1 ) then the correct scaling between mse and SNR is achieved when a → ∞, but the associated sequence of projection lattices converge to ZL−1 . The problem of finding good projections of the cubic lattice (and thus curves for this communication problem) can be independently formulated as the “fat strut” problem as follows. We want to find a point a ∈ ZL such that the cylinder anchored at the origin and a does not contain any other lattice point and has maximal volume. The Lifting Construction [26] gives a general solution for this problem. It is shown in [27] how to construct a sequences of lattices which are, up to equivalence relations, similar to projections of ZL and arbitrarily close to any target (L − 1)-dimensional lattice. By using layers of tori it is possible to generalize the construction in [22] as follows [23]. Let T = {T1 , . . . , TM } be a collection of M tori. For each one of these tori, consider curves of the form u ), s Tc (x) = Φc (x2πˆ
(8)
where C = diag(c1 , . . . , cN ), uˆ = u C = (c1 u1 , . . . , cL uN ), Φc is given by (1) and x ∈ [0, 1], PM Now let Len = j=1 Lenj , where Lenj is the length of s Tj . We split the unit interval [0, 1] into M pieces according to the length of each curve: [0, 1) = I1 ∪ I2 , . . . ∪ IM , where ! " Pk−1 Pk j=1 Lenj j=1 Lenj , for k = 1, . . . , M. , Ik = Len Len
(9)
and is represented in Figure 7. Finding a good collection of tori (i.e., such that each of them is separated at least a certain distance from each other) is related to finding a good spherical code of a given minimum distance, which can be approached through standard techniques (and even using layers of torus, as the construction presented in the previous section). On the other hand, finding good curves in each torus is equivalent to finding good projections of the rectangular lattice c1 Z ⊕ · · · ⊕ cL Z. In this case, it is possible to generalize the Lifting Construction and exhibit sequences of projections of c1 Z ⊕ · · · ⊕ cL Z converging to any (L − 1)-dimensional lattice, as in the later case. Through this, it is possible to meaningfully increase the length of the curves produced, while keeping a good distance between its laps, hence enhancing performance of the codes proposed in [22], comparing favorably also to other previous construction [28].
2πc1 1
1 IM
fk (x) Ik
2π
x 2πc2 u1
I2 I1
0
0
2πc1 u2
s(x)
Fig. 7. Encoding Process
C. A perspective on secrecy The codes proposed in the last section may also be used in terms of secrecy by considering, for instance, a wiretap channel with continuous input alphabet source. In this context a sender wishes to reliably transmit a real valued signal x ∈ R to a legitimate receiver while preventing an eavesdropper from correctly estimating x. Both channels (to the legitimate and to the eavesdropper) are AWGN channels, where noise has normal distribution with zero-mean and variances σb2 for the main channel (legitimate receiver) and σe2 for the wiretap
channel (eavesdropper), with σb2 > σe2 . Based on the output channel the legitimate user and the eavesdropper estimate, respectively, x ˆb and xˆe using some decoder that tries to minimize the mean square error (MSE) Fig. 8.
[6] G. D. F. Jr., “Geometrically uniform codes,” Information Theory, IEEE Transactions on, vol. 37, no. 5, pp. 1241 –1260, sep 1991.
nb legitimate
sender
x
encoder
⊕
y
decoderb
xˆb
[7] S. Costa, M. Muniz, E. Agustini, and R. Palazzo, “Graphs, tessellations, and perfect codes on flat tori,” Information Theory, IEEE Transactions on, vol. 50, no. 10, pp. 2363 – 2377, oct. 2004.
xˆe
[8] J. Stillwell, Geometry of surfaces, ser. Universitext. New York, Berlin, Heidelberg: Springer-Verlag, 1992. [Online]. Available: http://opac.inria.fr/record=b1129459
eavesdropper
⊕ z
decodere
ne Fig. 8. Wiretap channel
Three types of error decoding might occur when using curves on tori. Type I: the decoder can estimate correctly the torus and the curve fold but estimate incorrectly the transmitted point; type II: the decoder can estimate correctly the torus but estimate incorrectly the curve fold; type III: the decoder can estimate incorrectly the torus. Both errors of type I and type II induce small distortion (in terms of mse), while type III errors tend to induce large distortions. In order to deal with reliability and secrecy the goal is to design codes in which type I and type II errors are admissible for the legitimate receiver but not type III. On the other hand, we wish that, with high probability, the eavesdropper always decodes with errors of type III. This leads to adjusting properly the distance between flat tori, the small ball radius and also set the sTc mapping to account for secrecy. A good scheme should provide a small security gap [29] with high slope on the distortion curve (mse) as proposed in [30]. V.
C ONCLUSION
A commutative group code in R2L is always contained in a flat torus and it is also associated to a quotient of lattices in half of the dimension. On the other hand any sphere in even dimension can be foliated by flat tori. We present here an overview of these connections based on some previous works and discuss how they can be used to establish bounds and to construct discrete spherical codes, which are quasicommutative groups and to design spherical codes on flat torus layers for transmitting a continuous alphabet source. These continuous codes are related to lattice projections and may also be used in the context of physical security for a wiretap channel. Other perspectives include possible application by relating this topic to bandwidth compression [31]. R EFERENCES [1] [2]
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[5] I. Ingemarsson, “Group codes for the gaussian channel,” in Topics in Coding Theory, ser. Lecture Notes in Control and Information Sciences, G. Einarsson, T. Ericson, I. Ingemarsson, R. Johannesson, K. Zigangirov, and C.-E. Sundberg, Eds. Springer Berlin Heidelberg, 1989, vol. 128, pp. 73–108.
D. Slepian, “Group codes for the gaussian channel,” The Bell System Technical Journal, vol. 47, pp. 575 – 602, 1968. E. Biglieri and M. Elia, “Cyclic-group codes for the gaussian channel (corresp.),” Information Theory, IEEE Transactions on, vol. 22, no. 5, pp. 624 – 629, sep 1976. H.-A. Loeliger, “Signal sets matched to groups,” Information Theory, IEEE Transactions on, vol. 37, no. 6, pp. 1675 –1682, nov 1991. G. Caire and E. Biglieri, “Linear block codes over cyclic groups,” Information Theory, IEEE Transactions on, vol. 41, no. 5, pp. 1246 –1256, sep 1995.
[9] C. Torezzan, S. Costa, and V. Vaishampayan, “Spherical codes on torus layers,” in Proceedings of the IEEE International Symposium on Information Theory, June-July 2009, pp. 2033–2037. [10]
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H. Cohn and N. Elkies, “New upper bounds on sphere packings I,” Annals of mathematics, no. 157, pp. 689–714, 2003.
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W. Peterson, J. Nation, and M. Fossorier, “Reflection group codes and their decoding,” Information Theory, IEEE Transactions on, vol. 56, no. 12, pp. 6273 –6293, dec. 2010.
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G. F. Jr., “Coset codes. i. introduction and geometrical classification,” Information Theory, IEEE Transactions on, vol. 34, no. 5, pp. 1123 –1151, sep 1988.
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S. Costa, S. J.E., M. Alves, and C. T.B., “Circulant graphs and tesselations on flat tori,” Linear Algebra and Its Applications, vol. 432, pp. 369–382, 2010.
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A. A. El Gamal, L. A. Hemachandra, S. I., and V. K. Wei, “Using simulated anneling to design good codes,” IEEE Trans. Inform. Theory, vol. IT-33 no 1, pp. 116–123, 1987.
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J. Hamkins and K. Zeger, “Asymptotically dense spherical codes. i. wrapped spherical codes,” IEEE Trans. Inform. Theory, vol. 43, no. 6, pp. 1774–1785, Nov. 1997.
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——, “Asymptotically dense spherical codes .ii. laminated spherical codes,” IEEE Trans. Inform. Theory, vol. 43, no. 6, pp. 1786–1798, Nov. 1997.
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V. A. Vaishampayan and S. I. R. Costa, “Curves on a sphere, shift-map dynamics, and error control for continuous alphabet sources,” IEEE Trans. Inform. Theory, vol. 49, no. 7, pp. 1658–1672, 2003.
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A. Campello, C. Torezzan, and S. Costa, “Curves on torus layers and coding for continuous alphabet sources,” in Proceedings of the IEEE International Symposium on Information Theory, July 2012, pp. 2127– 2131.
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V. A. Vaishampayan, N. J. A. Sloane, and S. I. R. Costa, “Dynamical systems, curves and coding for continuous alphabet sources,” in Procedings International Telecommunications Symposium. Bangalore: ITW2002, 2002.
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VI.
A PPENDIX :
TABLE OF OPTIMUM COMMUTATIVE GROUP CODES IN R6 .
M 50 100 150 200 250
dmin 0.976312 0.804764 0.733971 0.673118 0.618034
c1 0.604 0.515 0.605 0.555 0.525
c2 0.506 0.684 0.516 0.619 0.525
c3 0.615 0.515 0.605 0.555 0.668
Group Z50 Z10 × Z10 Z150 Z200 Z5 × Z5 × Z10
300 350 400 450 500
0.585553 0.55558 0.54039 0.521353 0.504623
0.585 0.600 0.562 0.570 0.577
0.498 0.568 0.605 0.590 0.577
0.639 0.562 0.562 0.570 0.577
Z5 × Z60 Z350 Z20 × Z20 Z15 × Z30 Z5 × Z10 × Z10
550 600
0.483365 0.472569
0.588 0.549
0.605 0.630
0.534 0.549
Z550 Z2 × Z300
650
0.461475
0.601
0.525
0.601
Z5 × Z130
700 750 800 850 900 950 1000
0.445387 0.436651 0.42787 0.419318 0.413358 0.405197 0.397855
0.531 0.587 0.617 0.606 0.592 0.549 0.560
0.612 0.549 0.486 0.531 0.591 0.589 0.632
0.585 0.594 0.617 0.590 0.547 0.591 0.535
Z700 Z750 Z20 × Z40 Z850 Z3 × Z300 Z950 Z1000
TABLE II. S OME BEST COMMUTATIVE GROUP CODES IN R6 WITH 50 ≤ M ≤ 1000, INITIAL VECTOR c = (c1 , 0, c2 , 0, c3 , ), GENERATORS ( G EN ) GIVEN BY ROTATION BLOCKS WHERE bi1 , bi2 , bi3 AS IN P ROPOSITION 4 AND ISOMORPHIC GROUP (G ROUP ) COMPARED WITH THE T ORUS B OUND (B OUND ).
Gen (7,6, 34) (50, 10, 0), (30, 0, 10) (21, 25, 3) (28, 25, 4) (50, 0, 0), (50, 50, 0), (25, 25, 25) (0, 0, 60), (25, 30, 30) (123, 2, 33) (300, 40, 0), (60, 0, 20) (60, 0, 30), (15, 45, 45) (100, 0, 0), (50, 50, 0), (50, 0, 50) (189, 2, 47) (300, 0, 300), (384, 50, 12) (260, 520, 130), (55, 65, 15) (457, 664, 298) (187, 229, 560) (80,0,40),(20,80,60) (389,417,335) (0,300,0),(759,36,3) (33,23,98) (319,694,45)
Bound 1.09126 0.870236 0.76167 0.692764 0.643555 0.605911 0.57578 0.550878 0.529796 0.511615 0.4957 0.481599 0.468979 0.457586 0.447227 0.437747 0.429024 0.420958 0.413466 0.406481
Cristiano Torezzan
Antonio Campello
Vinay A. Vaishampayan
Institute of Mathematics School of Applied Sciences Institute of Mathematics AT&T Shannon Laboratory University of Campinas, Brazil University of Campinas, Brazil University of Campinas, Brazil Florham Park, NJ, USA [email protected] [email protected] [email protected] [email protected]
Abstract—The foliation of a sphere in an even number of dimensions by flat tori can be used to construct discrete spherical codes and also homogeneous curves for transmitting a continuous alphabet source over an AWGN channel. In both cases the performance of the code is related to the packing density of specific lattices and their orthogonal sublattices. In the continuous case the packing density of curves relies also on the search for projection lattices with good packing density. We present here a survey of this topic including some recent research results and perspectives. Keywords—Spherical codes, group codes, flat torus, lattices, Gaussian channel, codes on graphs, continuous alphabet source.
I.
I NTRODUCTION
Group codes as introduced by Slepian [1] and developed in subsequent articles [2], [3], [4], [5] are defined as finite sets on an n-dimensional sphere generated by the action of a group of orthogonal matrices. Geometrically uniform codes introduced by Forney [6] generalize this concept by considering also infinite sets of points in Euclidean space having a transitive symmetry group. We may say that the main course of what is presented here follows this concept in the context of metric spaces [7]: a signal set S ⊂ X is a geometrically uniform code if and only if for s, t in S there is an isometry f (depending on s, t) in X such that f (s) = t and f (S) = S. Geometrically uniform codes capture the highly desirable properties that come from homogeneity: a distance profile and error probability that is identical for every codeword, and Voronoi regions that are congruent to each other. One recurrent support metric space considered here is the n dimensional flat torus, obtained by identifying the opposite sides of a n dimensional box, more precisely defined as a quotient T = Rn /GT where GT is the group of translations generated by n independent vectors. A 2 − D flat torus can be visualized as a standard torus in the 3-dimensional space, but it can be distinguished from the latter by being locally flat i.e. like a piece of a plane. It can only be realized isometrically as a 2-dimensional surface in R4 which is contained in a 3-dimensional sphere. In this paper we present an overview of the strong connection between flat tori, lattices and group codes on a sphere, summarize some previous works and also point out recent results and perspectives of applications. Section II describes the foliation of a sphere in even dimensions, n = 2L, by L dimensional flat tori. Inequalities relating the distances on a flat torus in R2L and on its associated Work partially supported by FAPESP under grants 2009/18337-6, 2007/56052-8, 2012/09167-2 and CNPq 309561/2009-4
hyperbox in RL are also presented for later use. Section III is a strongly geometrical approach to commutative group codes presenting their connections with flat tori and quotients of lattices that allows the establishment of specific upper bounds for those codes. Some results on constructions which may approach those bounds and for optimal commutative group codes are discussed. Some remarks on commutative group codes considered on graphs are also included. Section IV-A summarizes a construction of spherical codes on layers of flat tori with some comparisons with well known spherical codes. In Section IV-B the homogeneous structure of flat tori and lattices come together again, now as a proposal for transmitting a continuous alphabet source over an AWGN channel. The search for projection lattices with good packing density is of significance here. Finally in Section V some concluding remarks including recent research and perspectives are drawn. II.
F LAT T ORI
The unit sphere S 2L−1 ⊂ R2L can be foliated by flat tori (Clifford Tori) as follows. For each unit vector c = (c1 , c2 , .., cL ) ∈ L X L−1 c2i = 1, and u = (u1 , u2 , . . . , uL ) ∈ RL , let S , ci > 0, i=1
Φc : RL → R2L be defined as u) = Φc (u
u1 u1 uL uL c1 cos( ), c1 sin( ), . . . , cL cos( ), cL sin( ) . c1 c1 cL cL (1)
The image of this periodic mapping Φc is the torus Tc , a flat L-dimensional surface on the unit sphere S 2L−1 . Tc is also the image of an L-dimensional box Pc , u ∈ RL ; 0 ≤ ui < 2πci }, 1 ≤ i ≤ L, Pc = {u
(2)
in which Pc is injective.
For c ∈ S L−1 and ci ≥ 0, if ci = 0 for some 1 ≤ i ≤ L, we may replace in (1) both coordinates related to ci by 0 and obtain a degenerated flat torus Tc , which is an embedding of a (L − k)-dimensional box in R2L , where k is the number of zero coordinates of c . The Gaussian curvature of a torus Tc is zero and Tc can be cut and flattened into the box, Pc , just as a cylinder in R3 can be cut and flattened into a 2-dimensional rectangle [8]. Since the inner product h∂Φc /∂ui , ∂Φc /∂uj i = δij , where δij is the Kronecker delta function, the application Φc is a local isometry, which means that any measure of length, area and volume up to dimension L − k on Tc is the same of the corresponding pre-image in the box Pc .
by u ), Φc (vv )|| = 2 ||Φc (u
S3
r X
c2i sin2 (
ui − vi ) 2ci
(4)
and it is bounded according to the next proposition [10]. Proposition 2. Let c = (c1 , c2 , .., cL ) ∈ S 2L−1 , ci > 0, cξ = u − v k for u , v ∈ Pc . Suppose 0 < ∆ ≤ min ci 6= 0, ∆ = ku
1≤i≤L
cξ /2, then
2∆ ≤ sin π
III.
∆ 2cξ
u) − Φc (vv )k ≤ 2cξ ≤ kΦc (u
sin ∆ 2 ≤∆ 2
C OMMUTATIVE G ROUP C ODES , F LAT T ORI AND L ATTICES
A. Commutative Group Codes
Pc
Let On be the multiplicative group of orthogonal n × n matrices and let Gn (M ) be the set of all order M commutative subgroups in On .
2πc2
A spherical commutative group code C is a set of M vectors which is the orbit of an initial vector u on the unit sphere S n−1 ⊂ Rn under the action of subgroup G of Gn (M ), u = {gu u, g ∈ G} . i.e. C := Gu
u
2πc1
The minimum distance in C is: d :=
c = (c1, c2)
min
x, y ∈ C x 6= y
||x − y|| =
min
gi 6= I ∈ G
||gi x − x||,
where ||.|| denotes the standard Euclidean norm. A canonical form for the a commutative group G ∈ Gn (M ) can be obtained from the following result.
S
Proposition 3. [[11],p 292] A commutative group G = {Oi }M i=1 of n × n orthogonal real matrices can be carried by one and the same real orthogonal transformation Q into the diagonal block canonical form QT Oi Q =
1
2πbi1 2πbiq ), . . . , R( ), µ2q+1 (i), . . . , µn (i)], (5) M M where bij are integers, the blocks R(a) are the 2-dimensional rotations cos(a) − sin(a) , R(a) = sin(a) cos(a) [R(
Fig. 1. Illustration of a torus layer Tc , c = (0.6, 0.8).
We say that the family of flat tori Tc and their degenerations, with c = (c1 , c2 , .., cL ), kcck = 1, ci ≥ 0, defined above is a foliation on the unit sphere of S 2L−1 ⊂ R2L . This means that any vector of S2N −1 belongs to one and only one of these flat tori. The following results [9] allow to relate the distances between two points in RL and their spherical image on flat tori in R2L Proposition 1. Let Tb and Tc be two flat tori, defined by unit vectors b and c with non negative coordinates. The minimum distance d(Tc , Tb ) between two points on these flat tori is !1/2 L X . (3) (ci − bi )2 d(Tc , Tb ) = kcc − b k =
and µl (i) = ±1 with l = 2q + 1, . . . , n. The next proposition [12] describes the geometric locus of a commutative group code. For even dimension this locus is always contained in a flat torus. Proposition 4. Every commutative group code of order M is, up to isometry, equal to a spherical code X whose initial vector is u = (u1 , . . . , un ) and its points have the form (R(ai1 )(u1 , u2 ), . . . , R(aiq )(u2q−1 , u2q ), µ2q+1 (i)u2q+1 , . . . , µn (i)un ),
1)
i=1
u ) and Φc (vv ) on the The distance between two points Φc (u same torus Tc , defined by a vector c = (c1 , · · · , cL ), is given
2πbij . Moreover, M If n = 2L, X is contained in the flat torus Tc , c = (c1 , . . . , cL ) where satisfies c 2i = u22i−1 + u22i . If n = 2L + 1 and X is substantial, X = X1 ∪ X2 , where Xi is contained in the plane Pi = {(x1 , . . . , x2L+1 ) ∈ R2L+1 ; x2L+1 = (−1)i un }.
where aij =
2)
Also, Xi is contained in the torus Tc of a sphere in R2m with radius (1−u2n )1/2 , where c 2i = u22i−1 +u22i . B. Lattice connections We say that a 2L-dimensional commutative group code is free from reflection blocks if its generator matrix group, considered as Proposition 3, satisfies 2L = 2q = n. By reflection blocks, we refer to the 2-dimensional blocks −1 0 , ± 0 1 which appear in the canonical form when 2q < n. Commutative group codes in even dimension, whose generator matrices are free from reflections blocks, are directly related to lattices. u we may For such commutative group codes C = Gu consider without loss of generality the initial vector as u = (c1 , 0, c2 , 0, . . . , cL , 0) where c = (c1 , c2 , .., cL ) is a unit vector. We also will consider here ci > 0 that is, codes that are not contained in a hyperplane of R2L . For the rotation angles aij = (2πbij )/M , where 1 ≤ i ≤ M , 1 ≤ j ≤ L as in proposition 4, let v i = (ai1 , . . . aiL ), 1 ≤ i ≤ M and the lattice Λ defined as the set of all integer combinations Q of v i . Note that Λ contains the orthogonal lattice Λc = (2πci )Z as a sublattice. The connection between these two lattices and u is given next [12]. the group code C = Gu
u with u = (c1 , 0, c2 , 0, . . . , cL , 0), Proposition 5. Let C = Gu c = (c1 , c2 , .., cL ), ||c|| = 1,ci > 0 be a commutative group code in R2L , free from reflection blocks. The inverse image Φc−1 by the torus mapping (1) is the lattice Λ defined as Λ above. Moreover the quotient of lattices is isomorphic to Λc the generator group G.
The last propositions combined with Proposition 2 allow to set bounds for commutative group codes via bounds for spherical packings on flat tori [12]. u of Proposition 7. Every commutative group code C = Gu order M in R2L free from 2 × 2 reflection blocks with initial vector u = (u1 , . . . , u2L ) and minimum distance d satisfies !L 2 1/2 2 ∆Guu π L ΠL π i=1 (u2i−1 + u2i ) M≤ ≤ ∆L , (arcsin d4 )L (arcsin d4 ).L1/2 where ∆Guu is the center density of the lattice Λ associated to the code and ∆L is the maximum center density of a lattice packing in RL . For general commutative group in R2L the lattice packing density can be replaced by the best periodical packing density in RL . Since any packing density in RL can be approached by periodical packing densities as remarked in [14], we can also replace ∆L in the last preposition for DL , by the best center packing density in RL [12]. u of Proposition 8. Every commutative group code C = Gu order M in R2L with initial vector u = (u1 , . . . , u2L ) and minimum distance d satisfies !L 2 2 1/2 π L ΠL DL π i=1 (u2i−1 + u2i ) M≤ ≤ DL , (arcsin d4 )L (arcsin d4 ).L1/2 where DL is the maximum center density of a spherical packing in RL . Bounds for commutative group codes in odd dimensions, n = 2L + 1, can also be obtained [12] by observing that those codes must lie on two parallel hyperplanes and is formed by two equivalent copies of commutative group codes in R2L .
7 14 21 3 10
7
2
17
12
22
17 24 6 13 20
9
4
19
14
24
2 9 16 23 5
6
1
16
11
21
12 19 1 8 15
3
8
13
18
23
22 4 11 18 0
0
5
10
15
20
Fig. 2. Pre-images Φc−1 of two cyclic group codes C = Gu of order M = 25 2π 2π7 in R√4 . On the the initial vector is u = √ left, G = h[R( 25 ), R( 25 ]i and 2π (1/ 2, 0, 1/ √ 2, 0). On the √ right side, G = h[R( 25 ), R( 2π10 ]i and initial 25 vector is u = ( 0.54915, 0, 0.45085, 0), which gives the best commutative group code of this order in R4 [13].
The inverse image through the torus mapping Φc of a commutative group code of order M generated by matrices which may contain 2 × 2 reflection blocks ( 2q < n in Proposition 4) is not always a quotient of lattices. However, from the L-periodicity of Φc in RL , we can write: Proposition 6. The inverse image through the torus mapping u of order M in R2L , Φc of any commutative group code Gu u = (u1 , . . . , u2L ) is a periodic distribution of M points in q L 2 2 the hyperbox Pc ⊂ R , ci = u2i−1 + u2i spanned by the lattice associated to this box.
The torus bounds given in propositions 7 and 8 are tight in the following sense: consider, for instance, the dual inequality of Proposition 8, ! L Y ci DL /M . d ≤ 2 sin i=1
For big M the distance d must be small (from Proposition 2) and the inverse image of the spherical cap of radius d in R2L centered in a point of Tc will be arbitrarily close to the ball of same radius in RL Fig. 3. This means that the best packing in the flat torus will be approached by the best packing in its pre-image in the box Pc and then the upper bounds of the above propositions will be approached. C. Approaching the bound: optimum commutative group codes For small distances δ or big M the search for good commutative group codes may rely on the search of orthogonal ˜ of a lattice Λ with good packing density. For sublattices Λ ˜ let b1 ≤ b2 ≤ . . . bL be length of the each such sublattice Λ P L 2 orthogonal basis vectors, b = b i=1 i and c = (c1 , . . . , cL ) ˜ The where ci = bi /b and the re-scaled lattices 1/bΛ and 1/bΛ. 1/bΛ commutative group code C associated to the quotient ˜ 1/bΛ on the flat torus Tc is a possible choice for a good code,
lattice ΛG (cc) where generator matrix T satisfies the following conditions: 1) 2) 3) 4)
Fig. 3. Inverse images of 4-dimensional balls of several radii though Φc , √ √ c = (2/ 5, 1/ 5). For a small radius it approaches a ball of the same radius in R2 .
particularly if Λ has the best packing density in L-dimensions. See [12] for examples in R6 of how the torus bounds can be approached when M increases. The algebraic group classification and matrix generators for the group code can be found via matrix reduction (Hermite and Smith Normal Forms) [13]. In what follows, C(M, n, d) denotes a code C in Rn with M points and minimum distance equal to d. A C(M, n, d) is said to be optimum if d is the largest minimum distance for a fixed M and n. As it is well known, the minimum distance of a group code C, generated by a finite group G, may vary significantly depending on the choice of the initial vector u. This problem still does not have a general solution, but has been studied in some important special cases, including reflection group codes [15] and permutation group codes [16]. Besides, Biglieri and Elia have shown in [2] that, for a fixed cyclic group code the problem can be formulated as a linear programming problem. They also discussed the efficiency of some of these codes and remarked on the hardness of obtaining the best cyclic group code for a given cardinality M and dimension n. In the search for the best commutative group code C(M, n, d), for fixed values of M and n we must first find the set Gn (M ) of all commutative groups in On of order M and then the best initial vector must be found for each one of those groups. An optimum code will be one which has the largest minimum distance in this set. The total number of Gn (M ) is related with the Euler number of divisors of M and is of order M/2 . It is worth to remark that even isomorphic groups n/2 must be considered, since the resulting minimal distance may vary depending on which representation in On is taken for each group, i.e. two isomorphic groups may generate two non isometric spherical codes, as illustrated in Fig. 2. Our approach to this problem in based on the association between commutative group codes and lattices described here. An important step of the algorithm derived in [13] is to reduce the number of cases to be analysed by discarding isometric codes. This is done via the following proposition. Proposition 9. Every commutative group code C(M, 2L, d), generated by a group G ∈ O2L free of 2 × 2 reflection blocks is isometric to a code obtained as the image by Φc of a
T is in the Hermite Normal Form; det(T ) = M L−1 ; There is a matrix W , with integer elements satisfying W T = M IL , where IL is the L × L identity matrix; The elements of the diagonal of T satisfy T (i, i) = M where ai is a divisor of M and (ai )i · ai (ai+1 · · · aL ) 6 M , ∀i = 1, . . . , L.
As an example of application of the above proposition let us consider M = 128. There are, up to isomorphism, only 4 abstract commutative groups or order 128: {Z128 , Z2 × Z64 , Z4 × Z32 , Z8 × Z16 }. However for n = 2L = {4, 6, 8} there are {2016, 41664, 635376} distinct representations of them in On . After discarding isometric codes by using Proposition 9 we must consider just {71, 2539, 55789} for n = 2L = {4, 6, 8}, respectively [13]. Then the initial vector problem will be solved only for those cases. D. Commutative group codes as codes on graphs Commutative group codes can also be viewed as a graph or a cosset code [17] on a flat torus with the graph distance (minimum number of edges from one vertex to another). They are also geometrically uniform in this context. This is the approach presented in [7]. As an example, consider the codes presented in Figure 2 where each edge of the flat torus box is subdivided into M = 25 segments with the underlined grid associated to this subdivision. Considering also the boundary identification, those grids define a graph on each flat torus 2 with vertices associated to the group Z25 . On the left we have the code generated by the element (b1, b2) = (1, 7), which is a perfect cyclic code in Z25 × Z25 of order M = 25 and minimum distance equal to 7, under the graph (or Lee) distance (a 3-error correcting code). On the right we have the code generated by (b1, b2) = (1, 10), which has minimum distance graph 5. Thus, viewed as graph codes, the code on left on Fig. 2 is better than the code on right in opposition to the performance of their images as spherical codes in R4 . Since both codes are isomorphic, as quotient of lattices, to the cyclic group Z25 they both are associated and present a more geometrical view of a circulant graph. (see Figure 4). This geometrical view may provide tools to analyse circulant and Cayley graphs which are used in parallel computing schemes [18]. IV.
S PHERICAL
CODES IN LAYERS OF TORI AND APPLICATIONS
A. Codes for the Gaussian Channel Although commutative group codes discussed in last section have applications based on their rich structure, those codes are not good in general concerning their the trade-off between distance and number of points, that is, the packing density on the sphere. Flat tori layers can be used to construct spherical codes which combine the good structure of commutative group codes in each layer with a better packing density. A Torus Layer Spherical Code (TLSC) can be generated by a finite set of orthogonal matrices and thus inherited group structure and
0
24
apple-peeling [19], a wrapped [20] and a laminated [21] codes, as illustrated in table I (see also [9] and [10]) for examples of TLSC in dimensions 6, 8 and 48.
1
23
2 3
22
4
21
5
20
19
6
18
7
d 0.5 0.4 0.3 0.2 0.1 0.01
TLSC(4,d) 172 308 798 2,718 22,406 2.27 ×107
TABLE I.
apple-peeling 136 268 676 2,348 19,364 1.97 ×107
wrapped * * * * 17,198 2.31 ×107
laminated * * * * 16,976 2.31 ×107
F OUR - DIMENSIONAL CODE SIZES AT VARIOUS MINIMUM DISTANCES . (*): UNKNOWN VALUES .
8
17 9
16 15
10
B. Curves on torus and continuous alphabet source
11
14 13
12
Fig. 4. The cyclic group code of Figure 2 left viewed as the circulant graph C25 (1, 7).
homogeneity allowing efficient storage and decoding process, which is attached to lattices in the half of the code √dimension. To design theses codes, given a distance d ∈ (0, 2], we first d
c4
d
c3
The homogeneous structure of flat tori and lattices can be used in a proposal for transmitting a continuous alphabet source over an AWGN channel [22], [23]. Consider the problem illustrated in Figure IV-B. A value x from a source with pdf having support [0, 1) is to be transmitted over a Gaussian channel of dimension N (or N orthogonal AWGN channels). The encoder will use a function s : [0, 1) → RN and then transmit the encoded value s (x) over the channel, such that the receiver will observe a noisy vector y = s(x) + z . The objective is to recover an estimate x of the transmitted value, attempting to minimize the mean square error (mse) ˆ 2 ]. If the N -dimensional Gaussian channel has E[(X − X)
d
c2 c1
d
d
d d
2πc12 2πc11
Pc
Fig. 5. Illustration of the construction of a four dimensional torus layer spherical code.
define a collection of tori in S 2L−1 such that the minimum distance between any two of these tori is at least d. This can be done (Proposition 1) by designing a spherical code in RL with minimum distance d and positive coordinates. Then, for each one of these tori, a finite set of points is chosen in RL such that the distance between any two points, when embedded in R2L by the standard parametrization (1), is greater than d, according to Proposition 2. This set of points may belong to a L−dimensional lattice, restricted to a hyperbox Pc (2), chosen to approach a good packing density in RL as described in Section III. The T LSC(2L, d) is the union of the commutative group codes associated to each one of the chosen tori. Figure 5 illustrates the construction of a T LSC(4, d). Those codes are homogeneous in each torus layers and have simplified coding and decoding processes (flat torus decoding in half of the dimension). For not very small distances (or non asymptotic context) a torus layer spherical code may have comparable performance to others well known spherical codes such as an
power P and variance σ 2 , then the average of the transmitted value should be no greater than P i.e. s should satisfy the 2 constraint E[kss(x)k ] ≤ P . This essentially means that the image of [0, 1) by the mapping s needs to be contained withing a sphere of radius P . On the other hand, it can be shown that for low noise, the mse of the scheme represented in Figure IV-B is given by Z 1 −2 2 2 ˆ ˆ 2 ], E[(X − X) ] ≈ σ f (x) ks˙ (x)k dx := Elow [(X − X) 0
(6) The quantity S(x) := ks˙ (x)k is called the stretch of the curve. Summarizing, we want to stretch the function as much as possible, but since s (x) is contained within a sphere, it has to be twisted, or folded. If the distance between the folds of the curve become too small, large errors will occur frequently when the noise is high, and the mse will approach Equation (6) only if the noise is sufficiently low. This is called the threshold effect. Figure 6 displays a picture of small and large errors. For the model described in Figure IV-B, the mse cannot decay faster than O(SN R−N ) [24]. It was shown in [22] that curves on the flat torus determined by the √ vector c = eˆ = (1/ L)(1, . . . , 1) can be used to obtain the correct scaling of the mse with the signal-to-noise ratio in
and consider the bijective mapping fk : Ik → [0, 1) Pk−1 x − j=1 Lenj /Len . fk (x) = Lenk /Len Then the full encoding map s can be defined by s (x) := s Tk (fk (x)), if x ∈ Ik .
Fig. 6. Illustration of small and large errors.
RN , N = 2L. Given a vector a ∈ ZL , the curves considered in [25] are of the type 2π s (x) = φeˆ √ a x(mod 1) , (7) L ax⌉. Due to Proposition (2), it can where a x(mod 1) = a x − ⌊a be shown that the small ball radius of s(x) is approximately the minimum distance of the lattice which is the projection of ZL onto the√subspace a ⊥ , while its stretch is constant and equal ak)2 . Thus, projections of the cubic lattice play to (2π/ L ka an important role on the design of such curves; the denser the projection lattice, the denser the curve on the sphere S 2L−1 . The result in [22] states that if a = (1, a, . . . , aL−1 ) then the correct scaling between mse and SNR is achieved when a → ∞, but the associated sequence of projection lattices converge to ZL−1 . The problem of finding good projections of the cubic lattice (and thus curves for this communication problem) can be independently formulated as the “fat strut” problem as follows. We want to find a point a ∈ ZL such that the cylinder anchored at the origin and a does not contain any other lattice point and has maximal volume. The Lifting Construction [26] gives a general solution for this problem. It is shown in [27] how to construct a sequences of lattices which are, up to equivalence relations, similar to projections of ZL and arbitrarily close to any target (L − 1)-dimensional lattice. By using layers of tori it is possible to generalize the construction in [22] as follows [23]. Let T = {T1 , . . . , TM } be a collection of M tori. For each one of these tori, consider curves of the form u ), s Tc (x) = Φc (x2πˆ
(8)
where C = diag(c1 , . . . , cN ), uˆ = u C = (c1 u1 , . . . , cL uN ), Φc is given by (1) and x ∈ [0, 1], PM Now let Len = j=1 Lenj , where Lenj is the length of s Tj . We split the unit interval [0, 1] into M pieces according to the length of each curve: [0, 1) = I1 ∪ I2 , . . . ∪ IM , where ! " Pk−1 Pk j=1 Lenj j=1 Lenj , for k = 1, . . . , M. , Ik = Len Len
(9)
and is represented in Figure 7. Finding a good collection of tori (i.e., such that each of them is separated at least a certain distance from each other) is related to finding a good spherical code of a given minimum distance, which can be approached through standard techniques (and even using layers of torus, as the construction presented in the previous section). On the other hand, finding good curves in each torus is equivalent to finding good projections of the rectangular lattice c1 Z ⊕ · · · ⊕ cL Z. In this case, it is possible to generalize the Lifting Construction and exhibit sequences of projections of c1 Z ⊕ · · · ⊕ cL Z converging to any (L − 1)-dimensional lattice, as in the later case. Through this, it is possible to meaningfully increase the length of the curves produced, while keeping a good distance between its laps, hence enhancing performance of the codes proposed in [22], comparing favorably also to other previous construction [28].
2πc1 1
1 IM
fk (x) Ik
2π
x 2πc2 u1
I2 I1
0
0
2πc1 u2
s(x)
Fig. 7. Encoding Process
C. A perspective on secrecy The codes proposed in the last section may also be used in terms of secrecy by considering, for instance, a wiretap channel with continuous input alphabet source. In this context a sender wishes to reliably transmit a real valued signal x ∈ R to a legitimate receiver while preventing an eavesdropper from correctly estimating x. Both channels (to the legitimate and to the eavesdropper) are AWGN channels, where noise has normal distribution with zero-mean and variances σb2 for the main channel (legitimate receiver) and σe2 for the wiretap
channel (eavesdropper), with σb2 > σe2 . Based on the output channel the legitimate user and the eavesdropper estimate, respectively, x ˆb and xˆe using some decoder that tries to minimize the mean square error (MSE) Fig. 8.
[6] G. D. F. Jr., “Geometrically uniform codes,” Information Theory, IEEE Transactions on, vol. 37, no. 5, pp. 1241 –1260, sep 1991.
nb legitimate
sender
x
encoder
⊕
y
decoderb
xˆb
[7] S. Costa, M. Muniz, E. Agustini, and R. Palazzo, “Graphs, tessellations, and perfect codes on flat tori,” Information Theory, IEEE Transactions on, vol. 50, no. 10, pp. 2363 – 2377, oct. 2004.
xˆe
[8] J. Stillwell, Geometry of surfaces, ser. Universitext. New York, Berlin, Heidelberg: Springer-Verlag, 1992. [Online]. Available: http://opac.inria.fr/record=b1129459
eavesdropper
⊕ z
decodere
ne Fig. 8. Wiretap channel
Three types of error decoding might occur when using curves on tori. Type I: the decoder can estimate correctly the torus and the curve fold but estimate incorrectly the transmitted point; type II: the decoder can estimate correctly the torus but estimate incorrectly the curve fold; type III: the decoder can estimate incorrectly the torus. Both errors of type I and type II induce small distortion (in terms of mse), while type III errors tend to induce large distortions. In order to deal with reliability and secrecy the goal is to design codes in which type I and type II errors are admissible for the legitimate receiver but not type III. On the other hand, we wish that, with high probability, the eavesdropper always decodes with errors of type III. This leads to adjusting properly the distance between flat tori, the small ball radius and also set the sTc mapping to account for secrecy. A good scheme should provide a small security gap [29] with high slope on the distortion curve (mse) as proposed in [30]. V.
C ONCLUSION
A commutative group code in R2L is always contained in a flat torus and it is also associated to a quotient of lattices in half of the dimension. On the other hand any sphere in even dimension can be foliated by flat tori. We present here an overview of these connections based on some previous works and discuss how they can be used to establish bounds and to construct discrete spherical codes, which are quasicommutative groups and to design spherical codes on flat torus layers for transmitting a continuous alphabet source. These continuous codes are related to lattice projections and may also be used in the context of physical security for a wiretap channel. Other perspectives include possible application by relating this topic to bandwidth compression [31]. R EFERENCES [1] [2]
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D. Slepian, “Group codes for the gaussian channel,” The Bell System Technical Journal, vol. 47, pp. 575 – 602, 1968. E. Biglieri and M. Elia, “Cyclic-group codes for the gaussian channel (corresp.),” Information Theory, IEEE Transactions on, vol. 22, no. 5, pp. 624 – 629, sep 1976. H.-A. Loeliger, “Signal sets matched to groups,” Information Theory, IEEE Transactions on, vol. 37, no. 6, pp. 1675 –1682, nov 1991. G. Caire and E. Biglieri, “Linear block codes over cyclic groups,” Information Theory, IEEE Transactions on, vol. 41, no. 5, pp. 1246 –1256, sep 1995.
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H. Cohn and N. Elkies, “New upper bounds on sphere packings I,” Annals of mathematics, no. 157, pp. 689–714, 2003.
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S. Costa, S. J.E., M. Alves, and C. T.B., “Circulant graphs and tesselations on flat tori,” Linear Algebra and Its Applications, vol. 432, pp. 369–382, 2010.
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A. A. El Gamal, L. A. Hemachandra, S. I., and V. K. Wei, “Using simulated anneling to design good codes,” IEEE Trans. Inform. Theory, vol. IT-33 no 1, pp. 116–123, 1987.
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J. Hamkins and K. Zeger, “Asymptotically dense spherical codes. i. wrapped spherical codes,” IEEE Trans. Inform. Theory, vol. 43, no. 6, pp. 1774–1785, Nov. 1997.
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——, “Asymptotically dense spherical codes .ii. laminated spherical codes,” IEEE Trans. Inform. Theory, vol. 43, no. 6, pp. 1786–1798, Nov. 1997.
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VI.
A PPENDIX :
TABLE OF OPTIMUM COMMUTATIVE GROUP CODES IN R6 .
M 50 100 150 200 250
dmin 0.976312 0.804764 0.733971 0.673118 0.618034
c1 0.604 0.515 0.605 0.555 0.525
c2 0.506 0.684 0.516 0.619 0.525
c3 0.615 0.515 0.605 0.555 0.668
Group Z50 Z10 × Z10 Z150 Z200 Z5 × Z5 × Z10
300 350 400 450 500
0.585553 0.55558 0.54039 0.521353 0.504623
0.585 0.600 0.562 0.570 0.577
0.498 0.568 0.605 0.590 0.577
0.639 0.562 0.562 0.570 0.577
Z5 × Z60 Z350 Z20 × Z20 Z15 × Z30 Z5 × Z10 × Z10
550 600
0.483365 0.472569
0.588 0.549
0.605 0.630
0.534 0.549
Z550 Z2 × Z300
650
0.461475
0.601
0.525
0.601
Z5 × Z130
700 750 800 850 900 950 1000
0.445387 0.436651 0.42787 0.419318 0.413358 0.405197 0.397855
0.531 0.587 0.617 0.606 0.592 0.549 0.560
0.612 0.549 0.486 0.531 0.591 0.589 0.632
0.585 0.594 0.617 0.590 0.547 0.591 0.535
Z700 Z750 Z20 × Z40 Z850 Z3 × Z300 Z950 Z1000
TABLE II. S OME BEST COMMUTATIVE GROUP CODES IN R6 WITH 50 ≤ M ≤ 1000, INITIAL VECTOR c = (c1 , 0, c2 , 0, c3 , ), GENERATORS ( G EN ) GIVEN BY ROTATION BLOCKS WHERE bi1 , bi2 , bi3 AS IN P ROPOSITION 4 AND ISOMORPHIC GROUP (G ROUP ) COMPARED WITH THE T ORUS B OUND (B OUND ).
Gen (7,6, 34) (50, 10, 0), (30, 0, 10) (21, 25, 3) (28, 25, 4) (50, 0, 0), (50, 50, 0), (25, 25, 25) (0, 0, 60), (25, 30, 30) (123, 2, 33) (300, 40, 0), (60, 0, 20) (60, 0, 30), (15, 45, 45) (100, 0, 0), (50, 50, 0), (50, 0, 50) (189, 2, 47) (300, 0, 300), (384, 50, 12) (260, 520, 130), (55, 65, 15) (457, 664, 298) (187, 229, 560) (80,0,40),(20,80,60) (389,417,335) (0,300,0),(759,36,3) (33,23,98) (319,694,45)
Bound 1.09126 0.870236 0.76167 0.692764 0.643555 0.605911 0.57578 0.550878 0.529796 0.511615 0.4957 0.481599 0.468979 0.457586 0.447227 0.437747 0.429024 0.420958 0.413466 0.406481
