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

A spherical code is a finite set X of points lying on the unit sphere of Rⁿ. For such a set, we define ρ(X) as the minimum of the squared distances ||x-y||², when x, y∈X and x≠y. Define R(ρ)=lim sup n→∞, ρ(X)=p log₂CardX/n. Chabauty in 1953 and Shannon in 1959 have given a lower bound for R(ρ), namely, R(ρ)>R_CS(ρ)=1-1/3log₂ρ(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_pol(ρ) is defined as above with the additional restriction that only families of codes with polynomial complexity of construction are considered. We prove R_pol(ρ)⩾R_CS(ρ)/2, if ρ⩽1.535. Denote by τ_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 θ_pol=lim sup(log₂τ_X(n))/n, when X ranges over all polynomially constructible families. We prove θ_pol⩾2/15=0.133···