Upper bounds on the minimum distance of spherical codes

We use linear programming techniques to obtain new upper bounds on the maximal squared minimum distance of spherical codes with fixed cardinality. Functions Qj(n, s) are introduced with the property that Qj(n, s) < 0 for some j > m iff the Levenshtein bound Lm(n, s) on A(n, s) = max{|W | : W is an (n, |W |, s) code} can be improved by a polynomial of degree at least m+1. General conditions on the existence of new bounds are presented. We prove that for fixed dimension n ≥ 5 there exist a constant k = k(n) such that all Levenshtein bounds Lm(n, s) for m ≥ 2k− 1 can be improved. An algorithm for obtaining new bounds is proposed and discussed.