Lattices and Spherical Codes Sueli I. R. Costa University of Campinas sueli@ime.unicamp.br London-ish Lattice Coding & Crypto Meeting January, 15th, 2018
Abstract Lattices in R n with orthogonal sublattices are associated with spheri- cal codes in R 2 n generated by a finite commutative group of orthog- onal matrices. They can also be used to construct homogeneous spherical curves for transmitting a continuous alphabet source over an AWGN channel. In both cases, the performance of the decod- ing process is related to the packing density of the lattices. In the continuous case, the “packing density” of these curves relies on the search for projection lattices with good packing density. A brief survey and recent developments on this topic is presented here.
Summary Spherical and Geometrically Uniform Codes; Flat Tori; Commutative Group Codes, Flat Tori and Lattices; Lattice bounds: Good and optimum commutative group codes; Spherical codes in layers of flat tori; Codes for continuous alphabet sources; Recent developments/perspectives;
General References T. Ericson, V. Zinoviev, Codes on Euclidean Spheres, North Holland, 2001; S. Costa, F. Oggier, A. Campello, J-C. Belfiore, E. Viterbo, Lattices Applied to Coding for Reliable and Secure Communications, Springer, 2018.
Spherical and Geometrically Uniform Codes Consider the sphere of radius a in R n , S n − 1 ( a ) = { x ∈ R n ; � x � = a } A spherical code is a finite set of M points on this sphere. Usually we consider only spherical codes on the sphere of radius one, S n − 1 = S n − 1 (1) and all the conclusions will be extended by similarity to a sphere of radius a .
Spherical and Geometrically Uniform Codes Two dual optimization (packing) problem, which have several applications in physics, chemistry, architecture and signal processing: Problem 1 : Given a dimension n and an integer number M > 0, to find a spherical code with M points such that the minimum distance between two points in the code is the largest possible. Problem 2 : Given a dimension n and a minimum distance d > 0, to find a spherical code with the biggest number M of points such that each two of them are at distance at least d .
Spherical and Geometrically Uniform Codes Codes which are solutions for one of these problems are called optimal spherical codes . In dimension 2: regular polygons. In dimesion 3: the solution of Problem 1 is only known for 1 ≤ M ≤ 12 and for M = 24. For M = 2 , 3 , 4 (antipodal points, equilateral triangle at the equator, regular tetrahedron). For M = 8: Figura: Antiprism with 8 vertices
M = 2 n in R n : biorthogonal code (permutations of (0 , 0 , ..., ± 1)); M = n + 1 in R n : simplex code ( y i permutations of n + n 2 (1 , 1 , . . . , 1 , − n ) ∈ R n +1 ). 1 √ � n +1 j =1 y ij = 0 (hyperplane), � n +1 j =1 y 2 ij = n + n 2 . Normalize and rotate → squared distance between two code words = 2 + 2 n ; Λ ⊂ R n a lattice, C ⊂ Λ → lattice vectors of a fixed norm.
Group Codes (Slepian 1958) Finite sets on an n -dimensional sphere generated by the action of a group of orthogonal matrices. Subsequent articles 70s – 90s: Biglieri, Elia, Loelinger, Caire, Ingemarsson Geometrically Uniform Codes (Forney 1991) For X a metric space, 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 . Highly desirable properties that come from homogeneity: the same distance profile, congruent Voronoi regions (same error transmission probability) for each codeword.
Examples of group codes in S 1 : A rotation group on the left, a group of reflexitions on the right (the initial vector matters).
Lattices in R n with orthogonal sublattices can be used to construct spherical codes in R 2 n generated by commutative groups of orthogonal matrices. Those codes will be contained on flat tori.
Flat Tori A 2-dimensional flat torus. For c = ( c 1 , c 2 ) with c 1 , c 2 positive numbers such that 2 = 1 , consider the map Φ c : R 2 → R 4 , defined as c 2 1 + c 2 Φ c ( u 1 , u 2 ) = ( c 1 cos( u 1 ) , c 1 sin( u 1 ) , c 2 cos( u 2 ) , c 2 sin( u 2 )) . c 1 c 1 c 2 c 2 This is a doubly periodic map having identical images in the translates of the rectangle [0 , 2 π c 1 ) × [0 , 2 π c 2 ) by vectors ( k 1 2 π c 1 , k 2 2 π c 2 ), k i integers. T c = Φ c ( R 2 ) = Φ c ([0 , 2 π c 1 ) × [0 , 2 π c 2 )) .
A view of the 2-dimensional flat torus which only can be realized in R 4 .
The unit sphere S 2 L − 1 ⊂ R 2 L can be foliated by flat tori (also L � c 2 called Clifford Tori): c = ( c 1 , c 2 , .., c L ) ∈ S L − 1 , c i > 0 , i = 1, i =1 and u = ( u 1 , u 2 , . . . , u L ) ∈ R L , let Φ c : R L → R 2 L be defined as � c 1 cos( u 1 ) , c 1 sin( u 1 ) , . . . , c L cos( u L ) , c L sin( u L � Φ c ( u ) = ) . c 1 c 1 c L c L (1) For P c = { u ∈ R L ; 0 ≤ u i < 2 π c i , 1 ≤ i ≤ L } . T c = φ c ( R l ) = φ c ( P ) ⊂ S 2 L − 1 . Any vector of S 2 L − 1 belongs to one and only one of these flat tori.
8 1.0 6 0.5 4 2 - 1.0 - 0.5 0.5 1.0 - 5 5 10 - 0.5 - 2 - 4 - 1.0 The tesselation of the plane associated to c =(0 . 8 , 0 . 6) ∈ S 1 , and a lattice Λ (black dots) which contains 2 π c 1 Z × 2 π c 2 Z as a rectangular sublattice. In this case φ c (Λ) is a spherical code in S 3 ⊂ R 4 with M = 8 .
Proposition Let T b and T c be two flat tori, defined by unit vectors b and c with non negative coordinates. The minimum distance d ( T c , T b ) between two points Φ c ( u ) and Φ c ( v ) on these flat tori is � L � 1 / 2 � ( c i − b i ) 2 d ( T c , T b ) = � c − b � = . (2) i =1 d c 4 d c 3 d c 2 c 1 d d d d 2 πc 1 2 P c 2 πc 1 1
Distance between two points Φ c ( u ) and Φ c ( v ) on the same torus: �� i sin 2 ( u i − v i c 2 || Φ c ( u ) − Φ c ( v ) || = 2 ) (3) 2 c i Proposition [VC03] Let c =( c 1 , c 2 , .., c L ) ∈ S L − 1 , c i > 0 , c ξ = min 1 ≤ i ≤ L c i � = 0 , ∆ = � u − v � for u , v ∈ P c . Suppose 0 < ∆ ≤ c ξ , then � ∆ 2∆ � 2 c ξ ≤ � Φ c ( u ) − Φ c ( v ) � ≤ 2 sin ∆ π ≤ sin 2 ≤ ∆ . 2 c ξ
Commutative Group Codes, Flat Tori and Lattices Lattice bounds: Good and optimum commutative group codes O n = the multiplicative group of orthogonal n × n matrices G n ( M ) = the set of all order M commutative subgroups in O n . 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 ⊂ R n by a given finite group G ∈ G n ( M ): C = G u = { g u , g ∈ G } . The minimum distance in C is: d = min || x − y || = g i � = I ∈ G || g i x − x || , min x , y ∈ C x � = y
A canonical form for a commutative group G ∈ G n ( M ). Proposition All the matrices O i of a commutative group G = { O i } M i =1 of n × n of orthogonal real matrices can simultaneously be put into a diagonal block canonical form through an orthogonal matrix Q: � � 2 π b i 1 � � 2 π b iq � � Q T O i Q = Rot , . . . , Rot , µ 2 q +1 ( i ) , . . . , µ n ( i ) , M M (4) where b ij are integers, the blocks Rot ( a ) are the ones associated with 2 -dimensional rotations by an angle of a radians: � cos( a ) � − sin( a ) Rot ( a ) = , sin( a ) cos( a ) and µ l ( i ) = ± 1 with l = 2 q + 1 , . . . , n.
The geometric locus of a commutative group code: Proposition Every commutative group code of order M is, up to isometry, equal to a spherical code X whose initial vector is u = ( u 1 , . . . , u n ) and its points have the form ( Rot ( a i 1 )( u 1 , u 2 ) , . . . , Rot ( a iq )( u 2 q − 1 , u 2 q ) , µ 2 q +1 ( i ) u 2 q +1 , . . . , µ n ( i ) u n ) , where a ij = 2 π b ij M . Moreover, 1. If n = 2 L, X is contained in the flat torus T c , c = ( c 1 , . . . , c L ) where c 2 i = u 2 2 i − 1 + u 2 2 i . 2. If n = 2 L + 1 and X is not contained in a hyperplane, X = X 1 ∪ X 2 , where X i is contained in the plane P i = { ( x 1 , . . . , x 2 L +1 ) ∈ R 2 L +1 ; x 2 L +1 = ( − 1) i u n } . Also, X i is contained in the torus T c of a sphere in R 2 L with radius (1 − u 2 n ) 1 / 2 , where c 2 i = u 2 2 i − 1 + u 2 2 i .
Lattice Connections A 2 L -dimensional commutative group code is free from reflection blocks if its generator matrix group satisfies 2 L = 2 q = n as in the Proposition (no blocks � − 1 0 � ± ) . 0 1 Commutative group codes in even dimension, whose generator matrices are free from reflections blocks, are directly related to lattices.
For such commutative group codes C = G u we may consider u = ( c 1 , 0 , c 2 , 0 , . . . , c L , 0) where c = ( c 1 , c 2 , .., c L ) ∈ S L − 1 , c i > 0 rotation angles a ij = (2 π b ij ) / M , 1 ≤ i ≤ M , 1 ≤ j ≤ L . v i = ( a i 1 , . . . a iL ), 1 ≤ i ≤ M and the lattice Λ with basis { v 1 , ..., v N } which has the rectangular sublattice Λ c = (2 π c 1 ) Z × (2 π c 1 ) Z × . . . × (2 π c L ) Z . Proposition [SC08] Let C = G u with u = ( c 1 , 0 , c 2 , 0 , . . . , c L , 0) , c = ( c 1 , c 2 , .., c L ) , || c || = 1 ,c i > 0 be a commutative group code in R 2 L , free from reflection blocks. The inverse image Φ − 1 by the c torus mapping (1) is the lattice Λ defined as above. Moreover the quotient of lattices Λ is isomorphic to the generator group G. Λ c
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