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http://dx.doi.org/10.4134/BKMS.b190325

ON GENERALIZATIONS OF SKEW QUASI-CYCLIC CODES  

Bedir, Sumeyra (Department of Mathematics Yildiz Technical University)
Gursoy, Fatmanur (Department of Mathematics Yildiz Technical University)
Siap, Irfan (Jacodesmath Institute)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.2, 2020 , pp. 459-479 More about this Journal
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.
Keywords
Skew cyclic codes; skew quasi-cyclic codes; generalized quasicyclic codes; multi-twisted codes; polycyclic codes;
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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   DOI
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   DOI
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   DOI
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   DOI
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   DOI
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   DOI
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   DOI
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   DOI
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   DOI
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   DOI
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   DOI
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   DOI
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   DOI
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.   DOI
19 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   DOI
20 N. Jacobson, Finite-Dimensional Division Algebras over Fields, Springer-Verlag, Berlin, 1996. https://doi.org/10.1007/978-3-642-02429-0
21 T. Koshy, Polynomial approach to quasi-cyclic codes, Bull. Calcutta Math. Soc. 69 (1977), no. 2, 51-59.
22 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   DOI
23 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   DOI
24 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   DOI
25 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   DOI
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 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   DOI
29 O. Ore, Theory of non-commutative polynomials, Ann. of Math. (2) 34 (1933), no. 3, 480-508. https://doi.org/10.2307/1968173   DOI
30 W. W. Peterson and E. J. Weldon, Jr., Error-Correcting Codes, second edition, The M.I.T. Press, Cambridge, MA, 1972.
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   DOI
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   DOI
33 E. J. Weldon, Jr., Long quasi-cyclic codes are good, IEEE Trans. Inform. Theory, IT-16 (1970), pp. 130.
34 I. Siap and N. Kulhan, The structure of generalized quasi cyclic codes, Appl. Math. E-Notes 5 (2005), 24-30.
35 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.