대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제51권4호
  • /
  • Pages.1175-1186
  • /
  • 2014
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

SELF-DUAL CODES AND FIXED-POINT-FREE PERMUTATIONS OF ORDER 2

  • Kim, Hyun Jin (Institute of Mathematical Sciences Ewha Womans University)
  • 투고 : 2013.10.06
  • 발행 : 2014.07.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

We construct new binary optimal self-dual codes of length 50. We develop a construction method for binary self-dual codes with a fixed-point-free automorphism of order 2. Using this method, we find new binary optimal self-dual codes of length 52. From these codes, we obtain Lee-optimal self-dual codes over the ring $\mathbb{F}_2+u\mathbb{F}_2$ of lengths 25 and 26.

키워드

참고문헌

  1. E. Bannai, M. Harada, A. Munemasa, T. Ibukiyamasa, and M. Oura, Type II codes over $F_2+_uF_2$ and applications to Hermitian modular forms, Abh. Math. Sem. Univ. Hamburg 73 (2003), 13-42. https://doi.org/10.1007/BF02941267
  2. A. Bonnecaze, P. Sole, and A. R. Calderbank, Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory 41 (1995), no. 2, 366-377. https://doi.org/10.1109/18.370138
  3. S. Buyuklieva, New binary extremal self-dual codes with lengths 50 and 52, Serdica Math. J. 12 (1999), no. 3, 185-190.
  4. S. Buyuklieva, A method for constucting self-dual codes with an automorphism of order 2, IEEE Trans. Inform. Theory 46 (2000), no. 2, 496-504. https://doi.org/10.1109/18.825812
  5. S. Buyuklieva and I. Boukliev, Extremal self-dual codes with an automorphism of order 2, IEEE Trans. Inform. Theory 44 (1998), no. 1, 323-328. https://doi.org/10.1109/18.651059
  6. S. Buyuklieva, M. Harada, and A. Munemasa, Restrictions on the weight enumerators of binary self-dual codes of length 4m, Proceedings of the International Workshop OCRT, White Lagoon, Bulgaria, 2007.
  7. J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory 36 (1990), no. 6, 1319-1333. https://doi.org/10.1109/18.59931
  8. S. T. Dougherty, P. Gaborit, M. Harada, A. Munemasa, and P. Sole, Type IV self-dual codes over rings, IEEE Trans. Inform. Theory 45 (1999), no. 7, 2345-2360. https://doi.org/10.1109/18.796375
  9. S. T. Dougherty, P. Gaborit, M. Harada, and P. Sole, Type II codes over $F_2+_uF_2$, IEEE Trans. Inform. Theory 45 (1999), no. 1, 32-45. https://doi.org/10.1109/18.746770
  10. S. T. Dougherty, T. A. Gulliver, and M. Harada, Extremal binary self-dual codes, IEEE Trans. Inform. Theory 43 (1997), no. 6, 2036-2047. https://doi.org/10.1109/18.641574
  11. A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. Sloane, and P. Sole, A linear construction for certain Kerdock and Preparata codes, Bull. Amer. Math. Soc. 29 (1993), no. 2, 218-222. https://doi.org/10.1090/S0273-0979-1993-00426-9
  12. M. Harada, Existence of new extremal doubly-even codes and extremal singly-even codes, Des. Codes Cryptogr. 8 (1996), no. 3, 273-283. https://doi.org/10.1023/A:1027303722125
  13. M. Harada, The existence of a self-dual [70, 35, 12] code and formally self-dual codes, Finite Fields Appl. 3 (1997), no. 2, 131-139. https://doi.org/10.1006/ffta.1996.0174
  14. M. Harada and A. Munemasa, Some restrictions on weight enumerators of singly even self-dual codes, IEEE Trans. Inform. Theory 52 (2006), no. 3, 1266-1269. https://doi.org/10.1109/TIT.2005.864416
  15. M. Harada, A. Munemasa, and V. Tonchev, A characterization of designs related to an extremal doubly-even self-dual code of length 48, Ann. Combin. 9 (2005), no. 2, 189-198. https://doi.org/10.1007/s00026-005-0250-x
  16. S. K. Houghten, C. W. H. Lam, L. H. Thiel, and J. A. Parker, The extended quadratic residue code is the only (48, 24, 12) self-dual doubly-even code, IEEE Trans. Inform. Theory 49 (2003), no. 1, 53-59. https://doi.org/10.1109/TIT.2002.806146
  17. W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Appl. 11 (2005), no. 3, 451-490. https://doi.org/10.1016/j.ffa.2005.05.012
  18. W. C. Huffman, On the decomposition of self-dual codes over M. Harada with an automorphism of odd prime order, Finite Fields Appl. 13 (2007), no. 3, 681-712. https://doi.org/10.1016/j.ffa.2006.02.003
  19. W. C. Huffman, Self-dual codes over M. Harada with an automorphism of odd order, Finite Fields Appl. 15 (2009), no. 3, 277-293. https://doi.org/10.1016/j.ffa.2007.07.003
  20. W. C. Huffman and V. Tonchev, The existence of extremal self-dual [50, 25, 10] codes and quasisymmetric 2 − (49, 9, 6), Des. Codes Cryptogr. 6 (1995), no. 2, 97-106. https://doi.org/10.1007/BF01398008
  21. W. C. Huffman and V. Tonchev, The [52, 26, 10] binary self-dual codes with an automorphism of order 7, in Proc. Optimal Codes and Related Topics, 127-136, Sozopol, Bulgaria, 1998.
  22. H. J. Kim, Lee-extremal self-dual codes over $F_2+_uF_2$ of lengths 23 and 24, Finite Fields Appl. 29 (2014), 18-33. https://doi.org/10.1016/j.ffa.2014.03.002
  23. H. J. Kim, H. Lee, J. B. Lee, and Y. Lee, Construction of self-dual codes with an automorphism of order p, Adv. Math. Commun. 5 (2011), no. 1, 23-36. https://doi.org/10.3934/amc.2011.5.23
  24. H. J. Kim and Y. Lee, Construction of extremal self-dual codes over $F_2+_uF_2$ with an automorphism of odd order, Finite Fields Appl. 18 (2012), no. 5, 971-992. https://doi.org/10.1016/j.ffa.2012.05.004
  25. H. J. Kim and Y. Lee, Hermitian self-dual codes over $\mathbb{F}_{2^{2m}}+u\mathbb{F}_{2^{2m}}$, Finite Fields Appl. 25 (2014), 106-131. https://doi.org/10.1016/j.ffa.2013.08.004
  26. J.-L. Kim, New extremal self-daul codes of length 36, 38, and 58, IEEE Trans. Inform. Theory 47 (2001), no. 1, 386-393. https://doi.org/10.1109/18.904540
  27. C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control 22 (1973), 188-200. https://doi.org/10.1016/S0019-9958(73)90273-8
  28. E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory 44 (1998), no. 1, 134-139. https://doi.org/10.1109/18.651000
  29. H.-P. Tsai, Existence of some extremal self-dual codes, IEEE Trans. Inform. Theory 38 (1992), no. 6, 1829-1833. https://doi.org/10.1109/18.165461
  30. N. Yankov, New optimal [52, 26, 10] self-dual codes, Des. Codes Cryptogr. 69 (2013), no. 2, 151-159. https://doi.org/10.1007/s10623-012-9639-9
  31. N. Yankov and R. Russeva, Binary self-dual codes of lengths 52 to 60 with an automor-phism of order 7 or 13, IEEE Trans. Inform. Theory 56 (2011), no. 11, 7498-7506.
  32. S. Zhang and S. Li, Some new extremal self-dual codes with lengths 42, 44, 52, and 58, Discrete Math. 238 (2001), no. 1-3, 147-150. https://doi.org/10.1016/S0012-365X(00)00420-9