By Topic

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

Sign In

Cookies must be enabled to login.After enabling cookies , please use refresh or reload or ctrl+f5 on the browser for the login options.

Formats Non-Member Member
$33 $13
Learn how you can qualify for the best price for this item!
Become an IEEE Member or Subscribe to
IEEE Xplore for exclusive pricing!
close button

puzzle piece

IEEE membership options for an individual and IEEE Xplore subscriptions for an organization offer the most affordable access to essential journal articles, conference papers, standards, eBooks, and eLearning courses.

Learn more about:

IEEE membership

IEEE Xplore subscriptions

2 Author(s)
G. Lachaud ; Lab. de Math. Discretes, CNRS, Marseille, France ; J. Stern

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···

Published in:

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