Uniqueness Of Codes Using Semidefinite Programming
Brouwer Andries E., Polak Sven C.. Arxiv 2017
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.
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