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 Q_j(n,s) are introduced with the property that Q_j(n,s)<0 for some j>m if and only if the Levenshtein bound L_m(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 exists a constant k=k(n) such that all Levenshtein bounds L_m(n, s) for m⩾2k-1 can be improved. An algorithm for obtaining new bounds is proposed and discussed