DOI QR코드

DOI QR Code

ON GENERALIZATIONS OF SKEW QUASI-CYCLIC CODES

  • Received : 2019.03.25
  • Accepted : 2019.11.19
  • Published : 2020.03.31

Abstract

In the last two decades, codes over noncommutative rings have been one of the main trends in coding theory. Due to the fact that noncommutativity brings many challenging problems in its nature, still there are many open problems to be addressed. In 2015, generator polynomial matrices and parity-check polynomial matrices of generalized quasi-cyclic (GQC) codes were investigated by Matsui. We extended these results to the noncommutative case. Exploring the dual structures of skew constacyclic codes, we present a direct way of obtaining parity-check polynomials of skew multi-twisted codes in terms of their generators. Further, we lay out the algebraic structures of skew multipolycyclic codes and their duals and we give some examples to illustrate the theorems.

References

  1. T. Abualrub, A. Ghrayeb, N. Aydin, and I. Siap, On the construction of skew quasicyclic codes, IEEE Trans. Inform. Theory 56 (2010), no. 5, 2081-2090. https://doi.org/10.1109/TIT.2010.2044062
  2. A. Alahamdi, S. Dougherty, A. Leroy, and P. Sole, On the duality and the direction of polycyclic codes, Adv. Math. Commun. 10 (2016), no. 4, 921-929. https://doi.org/10.3934/amc.2016049
  3. N. Aydin and A. Halilovic, A generalization of quasi-twisted codes: multi-twisted codes, Finite Fields Appl. 45 (2017), 96-106. https://doi.org/10.1016/j.ffa.2016.12.002
  4. N. Aydin, I. Siap, and D. Ray-Chaudhuri, The structure of 1-generator quasi-twisted codes and new linear codes, Des. Codes Cryptogr., 23 (2001), no. 3, 313-326.
  5. S. Bedir and I. Siap, Polycyclic codes over finite chain rings, International Conference on Coding and Cryptography, Algeria, 2015.
  6. W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235-265. https://doi.org/10.1006/ jsco.1996.0125
  7. D. Boucher, W. Geiselmann, and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engrg. Comm. Comput. 18 (2007), no. 4, 379-389. https://doi.org/10.1007/s00200-007-0043-z
  8. D. Boucher, P. Sole, and F. Ulmer, Skew constacyclic codes over Galois rings, Adv. Math. Commun. 2 (2008), no. 3, 273-292. https://doi.org/10.3934/amc.2008.2.273
  9. D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. Symbolic Comput. 44 (2009), no. 12, 1644-1656. https://doi.org/10.1016/j.jsc.2007.11.008
  10. D. Boucher and F. Ulmer, A note on the dual codes of module skew codes, in Cryptography and coding, 230-243, Lecture Notes in Comput. Sci., 7089, Springer, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-25516-8_14
  11. J. Conan and G. Seguin, Structural properties and enumeration of quasi-cyclic codes, Appl. Algebra Engrg. Comm. Comput. 4 (1993), no. 1, 25-39. https://doi.org/10.1007/BF01270398
  12. N. Fogarty and H. Gluesing-Luerssen, A circulant approach to skew-constacyclic codes, Finite Fields Appl. 35 (2015), 92-114. https://doi.org/10.1016/j.ffa.2015.03.008
  13. J. Gao, L. Shen, and F.-W. Fu, A Chinese remainder theorem approach to skew generalized quasi-cyclic codes over finite fields, Cryptogr. Commun. 8 (2016), no. 1, 51-66. https://doi.org/10.1007/s12095-015-0140-y
  14. M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, available at http://www.codetables.de.
  15. P. P. Greenough and R. Hill, Optimal ternary quasi-cyclic codes, Des. Codes Cryptogr. 2 (1992), no. 1, 81-91. https://doi.org/10.1007/BF00124211
  16. T. A. Gulliver and V. K. Bhargava, Nine good rate (m - 1)=pm quasi-cyclic codes, IEEE Trans. Inform. Theory 38 (1992), no. 4, 1366-1369. https://doi.org/10.1109/18.144718
  17. T. A. Gulliver and V. K. Bhargava, Some best rate 1/p and rate (p-1)/p systematic quasi-cyclic codes over GF(3) and GF(4), IEEE Trans. Inform. Theory 38 (1992), no. 4, 1369-1374. https://doi.org/10.1109/18.144719
  18. C. Guneri, F. Ozbudak, B. Ozkaya, E. Sacikara, Z. Sepasdar, and P. Sole, Structure and performance of generalized quasi-cyclic codes, Finite Fields Appl. 47 (2017), 183-202.
  19. N. Jacobson, Finite-Dimensional Division Algebras over Fields, Springer-Verlag, Berlin, 1996. https://doi.org/10.1007/978-3-642-02429-0
  20. S. Jitman, S. Ling, and P. Udomkavanich, Skew constacyclic codes over finite chain rings, Adv. Math. Commun. 6 (2012), no. 1, 39-63. https://doi.org/10.3934/amc.2012.6.39
  21. T. Koshy, Polynomial approach to quasi-cyclic codes, Bull. Calcutta Math. Soc. 69 (1977), no. 2, 51-59.
  22. K. Lally and P. Fitzpatrick, Algebraic structure of quasi-cyclic codes, Discrete Appl. Math. 111 (2001), no. 1-2, 157-175. https://doi.org/10.1016/S0166-218X(00)00350-4
  23. S. Ling and P. Sole, On the algebraic structure of quasi-cyclic codes. I. Finite fields, IEEE Trans. Inform. Theory 47 (2001), no. 7, 2751-2760. https://doi.org/10.1109/18.959257
  24. S. R. Lopez-Permouth, B. R. Parra-Avila, and S. Szabo, Dual generalizations of the concept of cyclicity of codes, Adv. Math. Commun. 3 (2009), no. 3, 227-234. https://doi.org/10.3934/amc.2009.3.227
  25. H. Matsui, On generator and parity-check polynomial matrices of generalized quasi-cyclic codes, Finite Fields Appl. 34 (2015), 280-304. https://doi.org/10.1016/j.ffa.2015.02.003
  26. M. Matsuoka, ${\theta}$-polycyclic codes and ${\theta}$-sequential codes over finite fields, Int. J. Algebra 5 (2011), no. 1-4, 65-70.
  27. B. R. McDonald, Finite Rings with Identity, Marcel Dekker, Inc., New York, 1974.
  28. O. Ore, Theory of non-commutative polynomials, Ann. of Math. (2) 34 (1933), no. 3, 480-508. https://doi.org/10.2307/1968173
  29. W. W. Peterson and E. J. Weldon, Jr., Error-Correcting Codes, second edition, The M.I.T. Press, Cambridge, MA, 1972.
  30. A. Sharma, V. Chauhan, and H. Singh, Multi-twisted codes over finite fields and their dual codes, Finite Fields Appl. 51 (2018), 270-297. https://doi.org/10.1016/j.ffa.2018.01.012
  31. I. Siap, T. Abualrub, N. Aydin, and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. J. Inf. Coding Theory 2 (2011), no. 1, 10-20. https://doi.org/10.1504/IJICOT.2011.044674
  32. I. Siap, N. Aydin, and D. K. Ray-Chaudhuri, New ternary quasi-cyclic codes with better minimum distances, IEEE Trans. Inform. Theory 46 (2000), no. 4, 1554-1558. https://doi.org/10.1109/18.850694
  33. I. Siap and N. Kulhan, The structure of generalized quasi cyclic codes, Appl. Math. E-Notes 5 (2005), 24-30.
  34. V. T. Van, H. Matsui, and S. Mita, Computation of Grobner basis for systematic encoding of generalized quasi-cyclic codes, IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences E92-A (2009), no.9, 2345-2359.
  35. E. J. Weldon, Jr., Long quasi-cyclic codes are good, IEEE Trans. Inform. Theory, IT-16 (1970), pp. 130.