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)>RCS(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 Rpol(ρ) is defined as above with the additional restriction that only families of codes with polynomial complexity of construction are considered. We prove Rpol(ρ)⩾RCS(ρ)/2, if ρ⩽1.535. Denote by τX(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 θpol=lim sup(log2τX(n))/n, when X ranges over all polynomially constructible families. We prove θpol⩾2/15=0.133···

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Information Theory, IEEE Transactions on  (Volume:40 ,  Issue: 4 )

Jul 1994