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 $$\mathcal{C}_k$$ be the collection of codes of cardinality at most k. Then $$A_q(n,d)$$ is at most the maximum value of $$\sum _{v\in [q]^n}x(\{v\})$$ , where x is a function $$\mathcal{C}_4\rightarrow {\mathbb {R}}_+$$ such that $$x(\emptyset )=1$$ and $$x(C)=\!0$$ if C has minimum distance less than d, and such that the $$\mathcal{C}_2\times \mathcal{C}_2$$ matrix $$(x(C\cup C'))_{C,C'\in \mathcal{C}_2}$$ 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 $$A_4(6,3)\le 176$$ , $$A_4(7,3)\le 596$$ , $$A_4(7,4)\le 155$$ , $$A_5(7,4)\le 489$$ , and $$A_5(7,5)\le 87$$ .