Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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MDS SELF-DUAL CODES OVER GALOIS RINGS WITH EVEN CHARACTERISTIC

  • Received : 2023.04.10
  • Accepted : 2023.08.21
  • Published : 2023.08.31
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Abstract

Let GR(2m, r) be a Galois ring with even characteristic. We are interested in the existence of MDS(Maximum Distance Separable) self-dual codes over GR(2m, r). In this paper, we prove that there exists an MDS self-dual code over GR(2m, r) with parameters [n, n/2, n/2 + 1] if (n - 1) | (2r - 1) and 8 | n.

Keywords

Acknowledgement

This paper was supported by the Education and Research Promotion Program of KOREATECH in 2023.

References

  1. Keith Conrad, Hensel's lemma, https://kconrad.math.uconn.edu/blurbs/gradnumthy/hensel.pdf
  2. S.T. Dougherty, K. Shiromoto, MDR Codes over Zk, IEEE-IT, 46 (2000), 265-269. https://doi.org/10.1109/18.817524
  3. X. Fang, K. Lebed, H. Liu, J. Luo, New MDS self-dual codes over finite fields of odd characteristic, Des. Codes Cryptogr., 88 (2020), 1127-1138. https://doi.org/10.1007/s10623-020-00734-x
  4. Fernando Q. Gouvea, p-adic Numbers An Introduction, Second Edition, Springer, 1997, Corrected 3rd printing 2003.
  5. M. Grassl, T.A. Gulliver, On self-dual MDS codes, In: Proceedings of ISIT (2008), 1954-1957.
  6. S. Han, MDS self-dual codes and antiorthogonal matrices over Galois rings, MDPI Information, 10 (2019), 1-12.
  7. S. Han, On the existence of MDS self-dual codes over finite chain rings, J. Chungcheong Math. Soc., 33 (2020), 255-270.
  8. S. Han, On the construction of MDS self-dual codes over Galois rings, Journal of Applied and Pure Mathematics, 4 (2022), 211-219. https://doi.org/10.23091/JAPM.2022.211
  9. W.C. Huffman, V.S. Pless, Fundamentals of Error-correcting Codes, Cambridge: Cambridge University Press, 2003.
  10. L. Jin and C. Xing, New MDS Self-Dual Codes From Generalized Reed?Solomon Codes, IEEE-IT, 63 (2017), 1434-1438. https://doi.org/10.1109/TIT.2016.2645759
  11. F.J. MacWilliams, N.J.A. Sloane, The Theory of Error-Correcting Codes, Amsterdam, The Netherlands: North-Holland, 1977.
  12. G.H. Norton, A. Salagean, On the structure of linear and cyclic codes over a finite chain ring, Appl. Algebra Engrg. Comm. Comput., 10 (2000), 489-506. https://doi.org/10.1007/PL00012382
  13. G.H. Norton, A. Salagean, On the key equation over a commutative ring, Designs, Codes and Cryptography, 20 (2000), 125-141. https://doi.org/10.1023/A:1008385407717
  14. G. Quintin, M. Barbier, C. Chabot, On Generalized Reed-Solomon Codes Over Commutative and Noncommutative Rings, IEEE-IT, 59 (2013), 5882-5897. https://doi.org/10.1109/TIT.2013.2264797
  15. Z.-X. Wan, Finite Fields and Galois Rings, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.
  16. H. Yan, A note on the constructions of MDS self-dual codes, Cryptogr. Commun., 11 (2019), 259-268. https://doi.org/10.1007/s12095-018-0288-3