Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

CYCLIC AND CONSTACYCLIC SELF-DUAL CODES OVER Rk

  • Received : 2015.09.17
  • Accepted : 2016.01.14
  • Published : 2017.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this work, we consider constacyclic and cyclic self-dual codes over the rings $R_k$. We start with theoretical existence results for constacyclic and cyclic self-dual codes of any length over $R_k$ and then construct cyclic self-dual codes over $R_1={\mathbb{F}}_2+u{\mathbb{F}}_2$ of even lengths from lifts of binary cyclic self-dual codes. We classify all free cyclic self-dual codes over $R_1$ of even lengths for which non-trivial such codes exist. In particular we demonstrate that our constructions provide a counter example to a claim made by Batoul et al. in [1] and we explain why their claim fails.

Keywords

References

  1. A. Batoul, K. Guenda, and T. A. Gulliver, On self-dual cyclic codes over finite chain rings, Des. Codes Cryptogr. 70 (2014), no. 3, 347-358. https://doi.org/10.1007/s10623-012-9696-0
  2. T. Blackford, Isodual constacyclic codes, Finite Fields Appl. 24 (2013), 29-44. https://doi.org/10.1016/j.ffa.2013.05.005
  3. A. Bonnecaze and P. Udaya, Cyclic codes and Self-dual codes over ${\mathbb{F}}_2$ + $u{\mathbb{F}}_2$, IEEE Trans. Inform. Theory 45 (1999), no. 4, 1250-1255. https://doi.org/10.1109/18.761278
  4. A. Bonnecaze and P. Udaya, Decoding of cyclic codes over ${\mathbb{F}}_2$+$u{\mathbb{F}}_2$, IEEE Trans. Inform. Theory 45 (1999), no. 6, 2148-2157. https://doi.org/10.1109/18.782165
  5. W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), 235-265. https://doi.org/10.1006/jsco.1996.0125
  6. B. Chan and H. Q. Dinh, A note on isodual constacyclic codes, Finite Fields Appl. 29 (2014), 243-246. https://doi.org/10.1016/j.ffa.2014.04.006
  7. S. T. Dougherty, P. Gaborit, M. Harada, and P. Sole, Type II codes over ${\mathbb{F}}_2$+$u{\mathbb{F}}_2$, IEEE Trans. Inform. Theory 45 (1999), no. 1, 32-45. https://doi.org/10.1109/18.746770
  8. S. T. Dougherty, S. Karadeniz, and B. Yildiz, Cyclic codes over $R_k$, Des. Codes Cryptogr. 63 (2012), no. 1, 113-126. https://doi.org/10.1007/s10623-011-9539-4
  9. S. T. Dougherty, B. Yildiz, and S. Karadeniz, Codes over $R_k$, Gray maps and their binary images, Finite Fields Appl. 17 (2011), no. 3, 205-219. https://doi.org/10.1016/j.ffa.2010.11.002
  10. S. T. Dougherty, B. Yildiz, and S. Karadeniz, Self-dual codes over $R_k$ and binary self-dual codes, Eur. J. Pure Appl. Math. 6 (2013), no. 1, 89-106.
  11. Y. Jia, S. Ling, and C. Xing, On self-dual cyclic codes over finite fields, IEEE Trans. Inform. Theory 57 (2011), no. 4, 2243-2251. https://doi.org/10.1109/TIT.2010.2092415
  12. S. Karadeniz, S. T. Dougherty, and B. Yildiz, Constructing formally self-dual codes over $R_k$, Discrete Appl. Math. 167 (2014), 188-196. https://doi.org/10.1016/j.dam.2013.11.017
  13. S. Karadeniz and B. Yildiz, (1 + v)-constacylic codes over ${\mathbb{F}}_2$ + $u{\mathbb{F}}_2$ + $v{\mathbb{F}}_2$ + $uv{\mathbb{F}}_2$, J. Franklin Inst. 348 (2011), no. 9, 2625-2632. https://doi.org/10.1016/j.jfranklin.2011.08.005
  14. S. Karadeniz, B. Yildiz, and N. Aydin, Extremal binary self-dual codes of lengths 64 and 66 from four-circulant constructions over codes ${\mathbb{F}}_2$ + $u{\mathbb{F}}_2$, FILOMAT to appear.
  15. V. Pless, P. Sole, and Z. Qian, Cyclic self-dual ${\mathbb{Z}}_4$-codes, Finite Fields Appl. 3 (1997), no. 1, 48-69. https://doi.org/10.1006/ffta.1996.0172
  16. N. J. A. Sloane and J. G. Thompson, Cyclic self-dual codes, IIEEE Trans. Inform. Theory 29 (1983), no. 3, 364-366. https://doi.org/10.1109/TIT.1983.1056682
  17. 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
  18. B. Yildiz and S. Karadeniz, Cyclic codes over ${\mathbb{F}}_2$ + $u{\mathbb{F}}_2$ + $v{\mathbb{F}}_2$ + $uv{\mathbb{F}}_2$, Des. Codes Cryptogr. 58 (2011), no. 3, 221-234. https://doi.org/10.1007/s10623-010-9399-3