We use linear programming techniques to obtain new upper bounds on the maximal squared minimum distance of spherical codes with fixed cardinality. Functions Q/sub j/(n,s) are introduced with the property that Q/sub j/(n,s)m if and only if the Levenshtein bound L/sub 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/spl ges/5 there exists a constant k=k(n) such that all Levenshtein bounds L/sub m/(n, s) for m/spl ges/2k-1 can be improved. An algorithm for obtaining new bounds is proposed and discussed.