Semidefinite programming, multivariate orthogonal polynomials, and codes in spherical caps

We apply the semidefinite programming approach developed in arxiv:math.MG/0608426 to obtain new upper bounds for codes in spherical caps. We compute new upper bounds for the one-sided kissing number in several dimensions where we in particular get a new tight bound in dimension 8. Furthermore we show how to use the SDP framework to get analytic bounds.Comment: 15 pages, (v2) referee comments and suggestions incorporate

Rigidity of spherical codes

A packing of spherical caps on the surface of a sphere (that is, a spherical code) is called rigid or jammed if it is isolated within the space of packings. In other words, aside from applying a global isometry, the packing cannot be deformed. In this paper, we systematically study the rigidity of spherical codes, particularly kissing configurations. One surprise is that the kissing configuration of the Coxeter-Todd lattice is not jammed, despite being locally jammed (each individual cap is held in place if its neighbors are fixed); in this respect, the Coxeter-Todd lattice is analogous to the face-centered cubic lattice in three dimensions. By contrast, we find that many other packings have jammed kissing configurations, including the Barnes-Wall lattice and all of the best kissing configurations known in four through twelve dimensions. Jamming seems to become much less common for large kissing configurations in higher dimensions, and in particular it fails for the best kissing configurations known in 25 through 31 dimensions. Motivated by this phenomenon, we find new kissing configurations in these dimensions, which improve on the records set in 1982 by the laminated lattices.Comment: 39 pages, 8 figure

On kissing numbers and spherical codes in high dimensions

We prove a lower bound of $\Omega (d^{3/2} \cdot (2/\sqrt{3})^d)$ on the kissing number in dimension $d$. This improves the classical lower bound of Chabauty, Shannon, and Wyner by a linear factor in the dimension. We obtain a similar linear factor improvement to the best known lower bound on the maximal size of a spherical code of acute angle $\theta$ in high dimensions

Asymptotic bounds for spherical codes

The set of all error-correcting codes C over a fixed finite alphabet F of cardinality q determines the set of code points in the unit square with coordinates (R(C), delta (C)):= (relative transmission rate, relative minimal distance). The central problem of the theory of such codes consists in maximizing simultaneously the transmission rate of the code and the relative minimum Hamming distance between two different code words. The classical approach to this problem explored in vast literature consists in the inventing explicit constructions of "good codes" and comparing new classes of codes with earlier ones. Less classical approach studies the geometry of the whole set of code points (R,delta) (with q fixed), at first independently of its computability properties, and only afterwords turning to the problems of computability, analogies with statistical physics etc. The main purpose of this article consists in extending this latter strategy to domain of spherical codes.Comment: 34 pages amstex, 3 figure

Upper bounds for packings of spheres of several radii

We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve bounds for the classical problem of packing identical spheres.Comment: 31 page

Constructive spherical codes on layers of flat tori

A new class of spherical codes is constructed by selecting a finite subset of flat tori from a foliation of the unit sphere S^{2L-1} of R^{2L} and designing a structured codebook on each torus layer. The resulting spherical code can be the image of a lattice restricted to a specific hyperbox in R^L in each layer. Group structure and homogeneity, useful for efficient storage and decoding, are inherited from the underlying lattice codebook. A systematic method for constructing such codes are presented and, as an example, the Leech lattice is used to construct a spherical code in R^{48}. Upper and lower bounds on the performance, the asymptotic packing density and a method for decoding are derived.Comment: 9 pages, 5 figures, submitted to IEEE Transactions on Information Theor