Uniqueness of codes using semidefinite programming

Abstract

For $n,d,w\in \mathbb{N}$, let $A(n,d,w)$ denote the maximum size of a binary code of word length $n$, minimum distance $d$ and constant weight $w$. Schrijver recently showed using semidefinite programming that $A(23,8,11)=1288$, and the second author that $A(22,8,11)=672$ and $A(22,8,10)=616$. Here we show uniqueness of the codes achieving these bounds. Let $A(n,d)$ denote the maximum size of a binary code of word length $n$ and minimum distance $d$. Gijswijt, Mittelmann and Schrijver showed that $A(20,8)=256$. We show that there are several nonisomorphic codes achieving this bound, and classify all such codes with all distances divisible by 4.

13 pages. Revisions have been made based on comments of the referees. Accepted for publication in Designs, Codes and Cryptography

References in corpus (2)

• Semidefinite bounds for nonbinary codes based on quadruples
• Semidefinite programming bounds for constant weight codes