Semidefinite bounds for nonbinary codes based on quadruples

For nonnegative integers q, n, d, let A_q(n, d) 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 A_q(n, d). For any k, let C_k be the collection of codes of cardinality at most k. Then A_q(n, d) is at most the maximum value of ∑_v∈[q]ⁿx({v}), where x is a function C₄→ R₊ such that x(∅) = 1 and x(C)=0 if C has minimum distance less than d, and such that the C₂× C₂ matrix (x(C∪C′))_C,C′∈C2 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 A₄(6,3) ≤ 176 , A₄(7,3) ≤ 596 , A₄(7,4) ≤ 155 , A₅(7,4) ≤ 489 , and A₅(7,5) ≤ 87.