Uniqueness of codes using semidefinite programming

TL;DR

This paper proves the uniqueness of certain optimal binary codes with fixed parameters using semidefinite programming and classifies all optimal codes with distances divisible by 4.

Contribution

It establishes the uniqueness of codes achieving specific bounds and classifies all optimal codes with distances divisible by 4.

Findings

Uniqueness of codes for A(23,8,11) and A(22,8,11)

Classification of all codes achieving A(20,8) with distances divisible by 4

Identification of multiple nonisomorphic codes at A(20,8)

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.