Semidefinite bounds for nonbinary codes based on quadruples

• Published in: arXiv.org
• Main Authors: Litjens, Bart, Polak, Sven, Schrijver, Alexander
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 08.02.2016
• Subjects: Codes; Integers; Mathematics - Combinatorics; Mathematics - Optimization and Control; Mathematics - Representation Theory; Polynomials; Semidefinite programming; Upper bounds
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1602.02531

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 \(\CC_4\to R_+\) such that \(x(\emptyset)=1\) and \(x(C)=0\) if \(C\) has minimum distance less than \(d\), and such that the \(\CC_2\times\CC_2\) 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 \(A_4(6,3)\leq 176\), \(A_4(7,4)\leq 155\), \(A_5(7,4)\leq 489\), and \(A_5(7,5)\leq 87\).