Uniqueness of codes using semidefinite programming

Andries E. Brouwer; Sven C. Polak

doi:10.1007/s10623-018-0589-8

Uniqueness of codes using semidefinite programming

Andries E. Brouwer, Sven C. Polak

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Abstract

For n, d, w ∈ 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 et al. 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.

Original language	English
Article number	8
Pages (from-to)	1881-1895
Number of pages	15
Journal	Designs Codes and Cryptography
Volume	87
Issue number	8
DOIs	https://doi.org/10.1007/s10623-018-0589-8 https://doi.org/10.1007/s10623-018-0589-8
Publication status	Published - 2019
Externally published	Yes

Keywords

Code
Binary code
Uniqueness
semidefinite programming
golay

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Cite this

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title = "Uniqueness of codes using semidefinite programming",

abstract = "For n, d, w ∈ 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 et al. 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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AU - Polak, Sven C.

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PY - 2019

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AB - For n, d, w ∈ 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 et al. 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.

KW - Code

KW - Binary code

KW - Uniqueness

KW - semidefinite programming

KW - golay

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JO - Designs Codes and Cryptography

JF - Designs Codes and Cryptography

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ER -

Uniqueness of codes using semidefinite programming

Abstract

Keywords

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