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Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 341–347 | Cite as

The sextuply shortened binary Golay code is optimal

  • Patric R. J. ÖstergårdEmail author
Article
  • 46 Downloads
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography

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(nd), it has been known that \(64 \le A(18,8) \le 68\) and \(128 \le A(19,8) \le 131\). 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.

Keywords

Classification Clique Double counting Error-correcting code Golay code 

Mathematics Subject Classification

94B25 94B65 90C27 

Notes

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Communications and NetworkingAalto University School of Electrical EngineeringAaltoFinland

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