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Self-Orthogonal 3-(56,12,65) Designs and Extremal Doubly-Even Self-Dual Codes of Length 56

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Abstract

In this paper, we show that the code generated by the rows of a block-point incidence matrix of a self-orthogonal 3-(56,12,65) design is a doubly-even self-dual code of length 56. As a consequence, it is shown that an extremal doubly-even self-dual code of length 56 is generated by the codewords of minimum weight. We also demonstrate that there are more than one thousand inequivalent extremal doubly-even self-dual [56,28,12] codes. This result shows that there are more than one thousand non-isomorphic self-orthogonal 3-(56,12,65) designs.

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References

  1. E. F. Assmus SuffixJr. H. F. Mattson SuffixJr. (1969) ArticleTitleNew 5-designs J. Combin. Theory. 6 122–151 Occurrence Handle42 #7528

    MathSciNet  Google Scholar 

  2. V. K. Bhargava G. Young A. K. Bhargava (1981) ArticleTitleA characterization of a (56, 28) extremal self-dual code IEEE Trans. Inform. Theory. 27 258–260 Occurrence Handle82i:94030 Occurrence Handle10.1109/TIT.1981.1056308

    Article  MathSciNet  Google Scholar 

  3. F. C. Bussemarker V. D. Tonchev (1989) ArticleTitleNew extremal doubly-even codes of length 56 derived from Hadamard matrices of order 28 Discrete Math. 76 45–49 Occurrence Handle90f:94055

    MathSciNet  Google Scholar 

  4. J. H. Conway N. J. A Sloane (1990) ArticleTitleA new upper bound on the minimal distance of self-dual codes IEEE Trans. Inform. Theory. 36 1319–1333 Occurrence Handle91m:94024

    MathSciNet  Google Scholar 

  5. M. Harada (1996) ArticleTitleExistence of new extremal doubly-even codes and extremal singly-even codes Des. Codes Cryptogr. 8 273–283 Occurrence Handle97g:94029 Occurrence Handle0981.94050

    MathSciNet  MATH  Google Scholar 

  6. M. Harada T. A. Gulliver H. Kaneta (1998) ArticleTitleClassification of extremal double-circulant self-dual codes of length up to 62 Discrete Math. 188 127–136 Occurrence Handle99c:94045 Occurrence Handle10.1016/S0012-365X(97)00250-1

    Article  MathSciNet  Google Scholar 

  7. M. Harada M. Kitazume A. Munemasa (2004) ArticleTitleOn a 5-design related to an extremal doubly even self-dual code of length 72 J. Combin. Theory Ser. A. 107 143–146 Occurrence Handle2005c:94070 Occurrence Handle10.1016/j.jcta.2004.03.005

    Article  MathSciNet  Google Scholar 

  8. H. Kimura (1994) ArticleTitleExtremal doubly even (56,28,12) codes and Hadamard matrices of order 28 Australas. J. Combin. 10 171–180

    Google Scholar 

  9. C. L. Mallows N. J. A. Sloane (1973) ArticleTitleAn upper bound for self-dual codes Inform. Control. 22 188–200 Occurrence Handle54 #2326 Occurrence Handle10.1016/S0019-9958(73)90273-8

    Article  MathSciNet  Google Scholar 

  10. V. D. Tonchev (1986) ArticleTitleA characterization of designs related to the Witt system S(5,8,24) Math. Z. 191 225–230 Occurrence Handle87g:05040 Occurrence Handle0563.05010 Occurrence Handle10.1007/BF01164026

    Article  MathSciNet  MATH  Google Scholar 

  11. V. D. Tonchev (1986) ArticleTitleQuasi-symmetric 2-(31,7,7) designs and a revision of Hamada’s conjecture J.Combin. Theory Ser. A. 42 104–110 Occurrence Handle87i:05043 Occurrence Handle0647.05010 Occurrence Handle10.1016/0097-3165(86)90009-9

    Article  MathSciNet  MATH  Google Scholar 

  12. V. Yorgov (1987) ArticleTitleA method for constructing inequivalent self-dual codes with applications to length 56 IEEE Trans. Inform Theory. 33 72–82 Occurrence Handle10.1109/TIT.1987.1057273

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Yamagata University, Yamagata, 990-8560, Japan

    Masaaki Harada

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  1. Masaaki Harada
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Correspondence to Masaaki Harada.

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J. D. Key

AMS Classification: 94B05, 05B05

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Harada, M. Self-Orthogonal 3-(56,12,65) Designs and Extremal Doubly-Even Self-Dual Codes of Length 56. Des Codes Crypt 38, 5–16 (2006). https://doi.org/10.1007/s10623-004-5657-6

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  • DOI: https://doi.org/10.1007/s10623-004-5657-6

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