The sextuply shortened binary Golay code is optimal

Patric R.J. Östergård

doi:10.1007/s10623-018-0532-z

The sextuply shortened binary Golay code is optimal

Patric R.J. Östergård^*

^*Corresponding author for this work

Information Theory

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Abstract

The maximum size of unrestricted binary three-error-correcting codes has been known up to the length of the binary Golay code, with two exceptions. Specifically, denoting the maximum size of an unrestricted binary code of length n and minimum distance d by A(n, d), it has been known that 64 ≤ A(18 , 8 ) ≤ 68 and 128 ≤ A(19 , 8 ) ≤ 131. In the current computer-aided study, it is shown that A(18 , 8 ) = 64 and A(19 , 8 ) = 128 , so an optimal code is obtained even after shortening the extended binary Golay code six times.

Original language	English
Pages (from-to)	341–347
Number of pages	7
Journal	Designs, Codes and Cryptography
Volume	87
Issue number	2-3
Early online date	13 Aug 2018
DOIs	https://doi.org/10.1007/s10623-018-0532-z
Publication status	Published - 15 Mar 2019
MoE publication type	A1 Journal article-refereed

Keywords

Classification
Clique
Double counting
Error-correcting code
Golay code
UPPER-BOUNDS
UNRESTRICTED CODES
ERROR-CORRECTING CODES

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title = "The sextuply shortened binary Golay code is optimal",

abstract = "The maximum size of unrestricted binary three-error-correcting codes has been known up to the length of the binary Golay code, with two exceptions. Specifically, denoting the maximum size of an unrestricted binary code of length n and minimum distance d by A(n, d), it has been known that 64 ≤ A(18 , 8 ) ≤ 68 and 128 ≤ A(19 , 8 ) ≤ 131. In the current computer-aided study, it is shown that A(18 , 8 ) = 64 and A(19 , 8 ) = 128 , so an optimal code is obtained even after shortening the extended binary Golay code six times.",

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AB - The maximum size of unrestricted binary three-error-correcting codes has been known up to the length of the binary Golay code, with two exceptions. Specifically, denoting the maximum size of an unrestricted binary code of length n and minimum distance d by A(n, d), it has been known that 64 ≤ A(18 , 8 ) ≤ 68 and 128 ≤ A(19 , 8 ) ≤ 131. In the current computer-aided study, it is shown that A(18 , 8 ) = 64 and A(19 , 8 ) = 128 , so an optimal code is obtained even after shortening the extended binary Golay code six times.

KW - Classification

KW - Clique

KW - Double counting

KW - Error-correcting code

KW - Golay code

KW - UPPER-BOUNDS

KW - UNRESTRICTED CODES

KW - ERROR-CORRECTING CODES

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ER -

The sextuply shortened binary Golay code is optimal

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Construction and Classification of Discrete Mathematic Structures

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The sextuply shortened binary Golay code is optimal

Abstract

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Construction and Classification of Discrete Mathematic Structures

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