Improved Delsarte Bounds via Extension of the Function Space

A spherical (n,N, α)-code is a set {x1, . . . ,xN} of unit vectors in R such that the pairwise angular distance betweeen the vectors is at least α. One tries to find codes which maximize N or α if the other two values are fixed. The kissing number problem asks for the maximum number k(n) of nonoverlapping unit balls touching a central unit ball in n-space. This corresponds to the special case of spherical codes that maximize N , for α = π 3 . In the early seventies Philippe Delsarte pioneered an approach that yields upper bounds on the cardinalities of binary codes and association schemes [3][4]. In 1977, Delsarte, Goethals and Seidel [5] adapted this approach to the case of spherical codes. The “Delsarte linear programming method” subsequently led to the exact resolution of the kissing number for dimensions 8 and 24, but also to the best upper bounds available today on kissing numbers, binary codes, and spherical codes (see Conway & Sloane [2]). Here we suggest and study strengthenings of the Delsarte method, for the setting of spherical codes and kissing numbers: We show that by extending the space of functions to be used, one can in some cases/dimensions improve the bounds that are achievable by the Delsarte method. Let X = (x1, . . . ,xN) ∈ R be an (n,N, α)-code, and let M = (xij) = (〈xi,xj〉) = XX ∈ R be the Gram matrix of scalar products of the xi. Then • xii = 1, while xij ≤ cosα for i 6= j, • M is symmetric and positive semidefinite, and • M has rank ≤ n.