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AN EFFICIENT CONSTRUCTION OF SELF-DUAL CODES

이윤진|김종락
대한수학회 (The Korean Mathematical Society)|2015.05|대한수학회보 (Bulletin of the KMS)|제52권 제3호|pp. 915-923

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참고문헌

키워드

linear codesself-dual codesbuilding-up construction
주제분류수학

초록

Self-dual codes have been actively studied because of their connections with other mathematical areas including t-designs, invariant theory, group theory, lattices, and modular forms. We presented the building-up construction for self-dual codes over GF(q) with q ≡ 1 (mod 4), and over other certain rings (see [19], [20]). Since then, the existence of the building-up construction for the open case over GF(q) with q = pr ≡ 3 (mod 4) with an odd prime p satisfying p ≡ 3 (mod 4) with r odd has not been solved. In this paper, we answer it positively by presenting the building-up construction explicitly. As examples, we present new optimal self-dual [16, 8, 7] codes over GF(7) and new self-dual codes over GF(7) with the best known parameters [24, 12, 9].

Abstract

Self-dual codes have been actively studied because of their connections with other mathematical areas including t-designs, invariant theory, group theory, lattices, and modular forms. We presented the building-up construction for self-dual codes over GF(q) with q ≡ 1 (mod 4), and over other certain rings (see [19], [20]). Since then, the existence of the building-up construction for the open case over GF(q) with q = pr ≡ 3 (mod 4) with an odd prime p satisfying p ≡ 3 (mod 4) with r odd has not been solved. In this paper, we answer it positively by presenting the building-up construction explicitly. As examples, we present new optimal self-dual [16, 8, 7] codes over GF(7) and new self-dual codes over GF(7) with the best known parameters [24, 12, 9].

참고문헌 (26)

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  3. "(ut)", Des. Codes Cryptogr, 74(2), pp.453~465 (2015)
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  9. T. A. Gulliver, "New MDS or near-MDS self-dual codes", IEEE Trans. Inform. Theory, 54(9), pp.4354~4360 (2008)
  10. S. Han, "A method for constructing self-dual codes over Z2m", Des. Codes Cryptogr, 75(2), pp.253~262 (2015)
  11. 한성휴, "CONSTRUCTION OF SELF-DUAL CODES OVER F2 + uF2", 대한수학회보, 49(1), pp.135~143 (2012), 대한수학회
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  16. K. F. Ireland, "A Classical Introduction to Modern Number Theory" (1982), Springer-Verlag
  17. J.-L. Kim, "New extremal self-dual codes of lengths 36, 38 and 58", IEEE Trans. Inform. Theory, 47, pp.386~393 (2001)
  19. J.-L. Kim, "Euclidean and Hermitian self-dual MDS codes over large finite fields", J. Combin. Theory Ser. A, 105(1), pp.79~95 (2004)
  20. J.-L. Kim, "Construction of MDS Self-dual codes over Galois rings", Des. Codes Cryptogr, 45(2), pp.247~258 (2007)
  21. J.-L. Kim, "Self-dual codes using the building-up construction", IEEE International Symposium on Information Theory, pp.2400~2402 (2009)
  22. H. Lee, "Construction of self-dual codes over finite rings Zpm", J. Combin. Theory Ser. A, 115(3), pp.407~422 (2008)
  23. J. S. Leon, "On ternary self-dual codes of length 24", IEEE Trans. Inform. Theory, 27(2), pp.176~180 (1981)
  24. V. Pless, "On the classification and enumeration of self-dual codes", J. Combin. Theory Ser. A, 18(3), pp.313~335 (1975)
  25. V. Pless, "Self-dual codes over GF(7)", IEEE Trans. Inform. Theory, 33(5), pp.723~727 (1987)
  26. E. Rains, "Handbook of Coding Theory" (1998), Elsevier

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