Semidefinite bounds for nonbinary codes based on quadruples

For nonnegative integers q, n, d, let [Formula: see text] denote the maximum cardinality of a code of length n over an alphabet [q] with q letters and with minimum distance at least d. We consider the following upper bound on [Formula: see text] . For any k, let [Formula: see text] be the collection of codes of cardinality at most k. Then [Formula: see text] is at most the maximum value of [Formula: see text] , where x is a function [Formula: see text] such that [Formula: see text] and [Formula: see text] if C has minimum distance less than d, and such that the [Formula: see text] matrix [Formula: see text] is positive semidefinite. By the symmetry of the problem, we can apply representation theory to reduce the problem to a semidefinite programming problem with order bounded by a polynomial in n. It yields the new upper bounds [Formula: see text] , [Formula: see text] , [Formula: see text] , [Formula: see text] , and [Formula: see text] .