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OPTIMAL LINEAR CODES OVER ℤm

  • 발행 : 2007.09.30

초록

We examine the main linear coding theory problem and study the structure of optimal linear codes over the ring ${\mathbb{Z}}_m$. We derive bounds on the maximum Hamming weight of these codes. We give bounds on the best linear codes over ${\mathbb{Z}}_8$ and ${\mathbb{Z}}_9$ of lengths up to 6. We determine the minimum distances of optimal linear codes over ${\mathbb{Z}}_4$ for lengths up to 7. Some examples of optimal codes are given.

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참고문헌

  1. T. Abualrub and R. Oehmke, On the generators of $Z_4$ cyclic codes of length $2^e$, IEEE Trans. Inform. Theory 49 (2003), no. 9, 2126-2133 https://doi.org/10.1109/TIT.2003.815763
  2. J. M. P. Balmaceda, A. L. Rowena, and F. R. Nemenzo, Mass formula for self-dual codes over $Z_{p2}$, Discrete Math. (to appear)
  3. T. Blackford, Cyclic codes over $Z_4$ of oddly even length, Discrete Appl. Math. 128 (2003), no. 1, 27-46 https://doi.org/10.1016/S0166-218X(02)00434-1
  4. A. R. Calderbank and N. J. A. Sloane, Modular and p-adic cyclic codes, Des. Codes Cryptogr. 6 (1995), no. 1, 21-35 https://doi.org/10.1007/BF01390768
  5. J. H. Conway and N. J. A. Sloane, Self-dual codes over the integers modulo 4, J. Combin, Theory Ser. A 62 (1993), no. 1, 30-45 https://doi.org/10.1016/0097-3165(93)90070-O
  6. S. T. Dougherty, T. A. Gulliver, and J. N. C. Wong, Self-dual codes over$Z_8\;and\;Z_9$, Des. Codes Cryptogr. 41 (2006), no. 3, 235-249 https://doi.org/10.1007/s10623-006-9000-2
  7. S. T. Dougherty, M. Harada, and P. Sole, Self-dual codes over rings and the Chinese remainder theorem, Hokkaido Math. J. 28 (1999), no. 2, 253-283 https://doi.org/10.14492/hokmj/1351001213
  8. S. T. Dougherty and S. Ling, Cyclic codes over $Z_4$ of even length, Des. Codes Cryptogr. 39 (2006), no. 2, 127-153 https://doi.org/10.1007/s10623-005-2773-x
  9. S. T. Dougherty, S. Y. Kim, and Y. H. Park, Lifted codes and their weight enumerators, Discrete Math. 305 (2005), no. 1-3, 123-135 https://doi.org/10.1016/j.disc.2005.08.004
  10. S. T. Dougherty and Y. H. Park, Codes over the p-adic integers, Des. Codes Cryptogr. 39 (2006), no. 1, 65-80 https://doi.org/10.1007/s10623-005-2542-x
  11. S. T. Dougherty and Y. H. Park, On modular cyclic codes, Finite Fields Appl. 13 (2007), no. 1, 31-57 https://doi.org/10.1016/j.ffa.2005.06.004
  12. 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
  13. S. T. Dougherty and K. Shiromoto, Maximum distance codes over rings of order 4, IEEE Trans. Inform. Theory 47 (2001), no. 1, 400-404 https://doi.org/10.1109/18.904544
  14. S. T. Dougherty and T. A. Szczepanski, Latin k-hypercubes, submitted
  15. J. Fields, P. Gaborit, J. S. Leon, and V. Pless, All self-dual $Z_4$ codes of length 15 or less are known, IEEE Trans. Inform. Theory 44 (1998), no. 1, 311-322 https://doi.org/10.1109/18.651058
  16. M. Greferath, G. McGuire, and M. O'Sullivan, On Plotkin-optimal codes over finite F'robenius rings, J. Algebra Appl. 5 (2006), no. 6, 799-815 https://doi.org/10.1142/S0219498806002022
  17. T. A. Gulliver and J. N. C. Wong, Classification of Optimal Linear $Z_4$ Rate 1/2 Codes of Length $\leq$ 8, submitted
  18. R. Hill, A First Course in Coding Theory, Oxford Applied Mathematics and Computing Science Series. The Clarendon Press, Oxford University Press, New York, 1986
  19. Y. H. Park, Modular Independence and Generator Matrices for Codes over $Z_m$, submitted https://doi.org/10.1007/s10623-008-9220-8
  20. W. C. Huffman and V. S. Pless, Fundamentals of Error-correcting Codes, Fundamentals of error-correcting codes. Cambridge University Press, Cambridge, 2003
  21. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, NorthHolland, Amsterdam, 1977
  22. G. H. Norton and A. Salagean, On the Hamming distance of linear codes over a finite chain ring, IEEE Trans. Inform. Theory 46 (2000), no. 3, 1060-1067 https://doi.org/10.1109/18.841186
  23. V. Pless, J. S. Leon, and J. Fields, All $Z_4$ codes of type II and length 16 are known, J. Combin. Theory Ser. A 78 (1997), no. 1,32-50 https://doi.org/10.1006/jcta.1996.2750
  24. K. Shiromoto and L. Storrne, A Griesmer Bound for Linear Codes over Finite QuasiFrobenius Rings, Discrete Appl, Math. 128 (2003), no. 1,263-274 https://doi.org/10.1016/S0166-218X(02)00450-X
  25. J. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121 (1999), no. 3, 555-575 https://doi.org/10.1353/ajm.1999.0024

피인용 문헌

  1. MDS and self-dual codes over rings vol.18, pp.6, 2012, https://doi.org/10.1016/j.ffa.2012.09.003
  2. The number of self-dual codes over $${Z_{p^3}}$$ vol.50, pp.3, 2009, https://doi.org/10.1007/s10623-008-9232-4
  3. MDS codes over finite principal ideal rings vol.50, pp.1, 2009, https://doi.org/10.1007/s10623-008-9215-5
  4. The classification of self-dual modular codes vol.17, pp.5, 2011, https://doi.org/10.1016/j.ffa.2011.02.010