Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

HIGHER WEIGHTS AND GENERALIZED MDS CODES

  • Received : 2009.02.04
  • Published : 2010.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

We study codes meeting a generalized version of the Singleton bound for higher weights. We show that some of the higher weight enumerators of these codes are uniquely determined. We give the higher weight enumerators for MDS codes, the Simplex codes, the Hamming codes, the first order Reed-Muller codes and their dual codes. For the putative [72, 36, 16] code we find the i-th higher weight enumerators for i = 12 to 36. Additionally, we give a version of the generalized Singleton bound for non-linear codes.

Keywords

References

  1. http://kutacc.kut.ac.kr/~sunghyu/data/hw/HW-P5-48-72.pdf
  2. http://kutacc.kut.ac.kr/~sunghyu/data/hw/HW-SHRM.pdf
  3. D. Britz, T. Britz, K. Shiromoto, and H. K. Sorensen, The higher weight enumerators of the doubly-even, self-dual [48, 24, 12] code, IEEE Trans. Inform. Theory 53 (2007), no. 7, 2567-2571. https://doi.org/10.1109/TIT.2007.899509
  4. J. Cannon and C. Playoust, An Introduction to Magma, University of Sydney, Sydney, Australia, 1994.
  5. S. T. Dougherty, T. A. Gulliver, and M. Oura, Higher weights and graded rings for binary self-dual codes, Discrete Appl. Math. 128 (2003), no. 1, 121-143. https://doi.org/10.1016/S0166-218X(02)00440-7
  6. S. T. Dougherty and R. Ramadurai, Higher weights of codes from projective planes and biplanes, Math. J. Okayama Univ. 49 (2007), 149-161.
  7. S. T. Dougherty and K. Shiromoto, MDR codes over $Z_k$, IEEE Trans. Inform. Theory 46 (2000), no. 1, 265-269. https://doi.org/10.1109/18.817524
  8. M. Grassl, Bounds on the minimum distance of linear codes, online available at http://www.codetables.de. Accessed on 2008-03-09.
  9. T. Helleseth, T. Klove, and J. Mykkeltveit, The weight distribution of irreducible cyclic codes with block length $n_1((q^l − 1)/N)$, Discrete Math. 18 (1977), no. 2, 179-211. https://doi.org/10.1016/0012-365X(77)90078-4
  10. H. Horimoto and K. Shiromoto, A Singleton bound for linear codes over quasi-Frobenius rings, Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, Hawaii (USA), 51-52 (1999).
  11. W. C. Huffman and V. S. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.
  12. T. Klove, Support weight distribution of linear codes, A collection of contributions in honour of Jack van Lint. Discrete Math. 106/107 (1992), 311-316.
  13. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes I, II, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
  14. G. McGuire and H. N. Ward, The weight enumerator of the code of the projective plane of order 5, Geom. Dedicata 73 (1998), no. 1, 63-77. https://doi.org/10.1023/A:1005088712217
  15. H. G. Schaathun, Duality and support weight distributions, IEEE Trans. Inform. Theory 50 (2004), no. 5, 862-867. https://doi.org/10.1109/TIT.2004.826673
  16. J. Simonis, The effective length of subcodes, Appl. Algebra Engrg. Comm. Comput. 5 (1994), no. 6, 371-377. https://doi.org/10.1007/BF01188748
  17. M. A. Tsfasman and S. G. Vladut¸, Geometric approach to higher weights, IEEE Trans. Inform. Theory 41 (1995), no. 6, part 1, 1564-1588. https://doi.org/10.1109/18.476213
  18. L. R. Vermani, Elements of Algebraic Coding Theory, Chapman and Hall Mathematics Series. Chapman and Hall, Ltd., London, 1996.
  19. V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory 37 (1991), no. 5, 1412-1418. https://doi.org/10.1109/18.133259