Polynomial-time construction of codes .II. spherical codes and the kissing number of spheres

Abstract

A spherical code is a finite set X of points lying on the unit sphere of R/sup n/. For such a set, we define /spl rho/(X) as the minimum of the squared distances /spl par/x-y/spl par//sup 2/, when x, y/spl isin/X and x/spl ne/y. Define R(/spl rho/)=lim sup n/spl rarr//spl infin/, /spl rho/(X)=p log/sub 2/CardX/n. Chabauty in 1953 and Shannon in 1959 have given a lower bound for R(/spl rho/), namely, R(/spl rho/)>R/sub CS/(/spl rho/)=1-1/3log/sub 2//spl rho/(4-p). The complexity of construction of the spherical codes used in order to get this bound is doubly exponential. The polynomially constructible spherical bound R/sub pol/(/spl rho/) is defined as above with the additional restriction that only families of codes with polynomial complexity of construction are considered. We prove R/sub pol/(/spl rho/)/spl ges/R/sub CS/(/spl rho/)/2, if /spl rho//spl les/1.535. Denote by /spl tau//sub X/(n) the number of spheres of equal radius that touch one sphere in the n-dimensional space given by some explicit family X, that is, a family of arrangements of spheres). The asymptotic polynomially constructible kissing number is /spl theta//sub pol/=lim sup(log/sub 2//spl tau//sub X/(n))/n, when X ranges over all polynomially constructible families. We prove /spl theta//sub pol//spl ges/2/15=0.133/spl middot//spl middot//spl middot/. >