Skew cyclic codes over Z4+vZ4 with derivation: structural properties and computational results

Document Type : Original paper

Authors

1 Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung, 40132, Indonesia

2 School of Mathematics and Statistics, UNSW, Sydney, Australia

Abstract

In this work, we study a class of skew cyclic codes over the ring R:=Z4+vZ4, where v2=v, with an automorphism θ and a derivation Δθ, namely codes as modules over a skew polynomial ring R[x;θ,Δθ], whose multiplication is defined using an automorphism θ and a derivation Δθ. We investigate the structures of a skew polynomial ring R[x;θ,Δθ]. We define Δθ-cyclic codes as a generalization of the notion of cyclic codes. The properties of Δθ-cyclic codes as well as dual Δθ-cyclic codes are derived. As an application, some new linear codes over Z4 with good parameters are obtained by Plotkin sum construction, also via a Gray map as well as residue and torsion codes of these codes.

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References

[1] T. Abualrub, N. Aydin, and P. Seneviratne, On θ-cyclic codes over F2+vF2, Australas. J. Combin 54, no. 2.  115–126.
[2] N. Aydin and T. Asamov, The database of Z4 codes, available at http://quantumcodes.info/Z4 (accessed at January 5, 2024).
[3] R.K. Bandi and M. Bhaintwal, Codes over Z4+vZ4, 2014 International Conference on Advances in Computing, Communications and Informatics (ICACCI), 2014, pp. 422–427. https://doi.org/10.1109/ICACCI.2014.6968489
[4] N. Benbelkacem, M.F. Ezerman, T. Abualrub, N. Aydin, and A. Batoul, Skew cyclic codes over F4R, J. Algebra Appl. 21 (2022), no. 4, Article ID: 2250065.  https://doi.org/10.1142/S0219498822500657
[5] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system I: The user language, J. Symb. Comput. 24 (1997), no. 3-4, 235–265.  https://doi.org/10.1006/jsco.1996.0125
[6] 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
[7] D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings, Cryptography and Coding: 12th IMA International Conference, Cryptography and Coding 2009, Cirencester, UK, December 15-17, 2009. Proceedings 12 (M.G.
Parker, ed.), Springer, 2009, pp. 38–55.
[8] D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. Symb. Comput. 44 (2009), no. 12, 1644–1656. https://doi.org/10.1016/j.jsc.2007.11.008
[9] D. Boucher and F. Ulmer, Linear codes using skew polynomials with automorphisms and derivations, Des. Codes, Cryptogr. 70 (2014), no. 3, 405–431.  https://doi.org/10.1007/s10623-012-9704-4
[10] A.R. Calderbank, A.R. Hammons Jr, P.V. Kumar, N.J.A. Sloane, and P. Solé, The Z4-linearity of kerdock, preparata, goethals and related codes, IEEE Trans. Inf. Theory 40 (1994), no. 2, 301–319.
[11] J. Gao, F.W. Fu, and Y. Gao, Some classes of linear codes over Z4+vZ4 and their applications to construct good and new Z4 linear codes, Appl. Algebra Engrg. Comm. Comput. 28 (2016), no. 2, 131–153. https://doi.org/10.1007/s00200-016-0300-0
[12] F. Gursoy, I. Siap, and B. Yildiz, Construction of skew cyclic codes over Fq+vFq, Adva. Math. Commun. 8 (2014), no. 3, 313–322.  https://doi.org/10.3934/amc.2014.8.313
[13] W.C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge university press, New York, 2003.
[14] A. B. Irwansyah and D. Suprijanto, Structure of linear codes over the ring Bk, J. Appl. Math. Comput. 58 (2018), no. 1–2, 755–775.  https://doi.org/10.1007/s12190-018-1165-0
[15] A.B. Irwansyah, S.T. Dougherty, A. Muchlis, I. Muchtadi-Alamsyah, P. Solé, D. Suprijanto, and O. Yemen, ΘS-cyclic codes over Ak, Int. J. Comput. Math. Comput. Syst. Theory 1 (2016), no. 1, 14–31.
[16] A.B. Irwansyah, I. Muchtadi-Alamsyah, A. Muchlis, and D. Suprijanto, Skew-cyclic codes over Bk, J. Appl. Math. Comput. 57 (2018), no. 1-2, 69–84.  https://doi.org/10.1007/s12190-017-1095-2
[17] J. Liu and H. Liu, Construction of cyclic DNA codes over the ring Z4+vZ4, IEEE Access 8 (2020), 111200–111207.  https://doi.org/10.1109/ACCESS.2020.3001283
[18] B.R. McDonald, Finite Rings with Identity, Marcel Dekker Inc., New York, 1974.
[19] O. Ore, Theory of non-commutative polynomials, Annals. Math. 34 (1933), no. 3, 480–508. https://doi.org/10.2307/1968173
[20] S. Patel and O. Prakash, (θ,δθ)-Cyclic codes over Fq[u,v]/⟨u2−u,v2−vuv−vu⟩, Des. Codes Cryptogr. 90 (2021), no. 11, 2763–2781.  https://doi.org/10.1007/s10623-021-00964-7
[21] E. Prange, Cyclic Error-correcting Codes in Two Symbols, AFCRC-TN, Air Force Cambridge Research Center, 1957.
[22] A. Sharma and M. Bhaintwal, A class of skew-cyclic codes over Z4+uZ4 with derivation, Adv. Math. Commun. 12, no. 4, 723–739.  https://doi.org/10.3934/amc.2018043
[23] H.C. Tang and D. Suprijanto, New optimal linear codes over Z4, Bull. Aust. Math. Soc. 107 (2023), no. 1, 158–169.  https://doi.org/10.1017/S0004972722000399
[24] Z.X. Wan, Quaternary Codes, vol. 8, World Scientific, Singapore, 1997.
Communications in Combinatorics and Optimization
Volume 10, Issue 3
September 2025
Pages 497-517

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