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Designs, Codes and Cryptography

, Volume 74, Issue 3, pp 673–680 | Cite as

New extremal binary self-dual codes of length 64 from \(R_3\)-lifts of the extended binary Hamming code

  • Suat Karadeniz
  • Bahattin YildizEmail author
Article

Abstract

In this paper, we use the graded ring construction to lift the extended binary Hamming code of length 8 to \(R_k\). Using this method we construct self-dual codes over \(R_3\) of length 8 whose Gray images are self-dual binary codes of length 64. In this way, we obtain twenty six non-equivalent extremal binary Type I self-dual codes of length 64, ten of which have weight enumerators that were not previously known to exist. The new codes that we found have \(\beta = 1, 5, 13, 17, 21, 25, 29, 33, 41\) and 52 in \(W_{64,2}\) and they all have automorphism groups of size 8.

Keywords

Lee weight Gray maps Extremal self-dual codes Projections Lifts  Codes over rings 

Mathematics Subject Classification

94B05 94B65 

Notes

Acknowledgments

The authors would like to thank the anonymous referees for their valuable remarks and suggestions that improved the presentation of the paper considerably.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsFatih UniversityIstanbulTurkey

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