Semidefinite bounds for nonbinary codes based on quadruples - Litjens, Bart; Polak, Sven; Schrijver, Alexander (arXiv.org, 2016)
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 \(\CC_k\) 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 \(\CC_4→ R_+\) such that \(x(\emptyset)=1\) and \(x(C)=0\) if \(C\) has minimum distance less than \(d\), and such that the \(\CC_2×\CC_2\) matrix \((x(C\cup C'))_{C,C'\in\CC_2}\) 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 \(A_4(6,3)≤ 176\), \(A_4(7,4)≤ 155\), \(A_5(7,4)≤ 489\), and \(A_5(7,5)≤ 87\).