Abstract
In this paper, we give conditions on the existence of Euclidean self-dual skew cyclic and skew negacyclic codes over the finite chain ring \(\mathbb {F}_q+u\mathbb {F}_q\). We also extend an algorithm of Boucher and Ulmer [J. Symbolic Comput. 60, 2014] to construct self-dual skew cyclic and skew negacyclic codes based on the least common left multiples of non-commutative polynomials over \(\mathbb {F}_q+u\mathbb {F}_q\). Furthermore, we give conditions on the existence of LCD skew cyclic and skew negacyclic codes. Detailed examples are given which were obtained with the aid of the Magma Computational Algebra System Bosma et al. (J Symbolic Comput 24(3–4):235–265, 1997).
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References
Ashraf M, Mohammad G (2015) On skew cyclic codes over a semi-local ring, Discrete Math., Algorithms Applic 7(3)
Batoul A, Guenda K, Gulliver TA, Aydin N (2017) Constacyclic codes over finite principal ideal rings, Codes, vol 10194. Lecture Notes in Computer Science. Springer, Berlin, pp 161–175
Bennenni N, Guenda K, Mesnager S (2017) DNA cyclic codes over rings. Adv Math Commun 11(1):83–98
Bosma W, Cannon J, Playoust C (1997) The Magma algebra system. I. The user language. J Symbolic Comput 24(3–4):235–265
Boucher D, Ulmer F (2009) Codes as modules over skew polynomial rings. Lecture Notes Comput Sci 5921:38–55
Boucher D (2015) A note on the existence of self-dual skew codes over finite fields, vol 9084. Lecture Notes in Computer Science. Springer, Berlin, pp 228–239
Boucher D, Ulmer F (2011) A note on the dual codes of module skew codes, vol 7089. Lecture Notes in Computer Science. Springer, Berlin, pp 230–243
Boucher D (2016) Construction and number of self-dual skew codes over \(\mathbb{F}_{p^{2}}\). Adv Math Commun 10(4):765–795
Boucher D, Ulmer F (2014) Self-dual skew codes and factorization of skew polynomials. J Symbolic Comput 60:47–61
Boucher D, Solé P, Ulmer F (2008) Skew constacyclic code over Galois rings. Adv Math Commun 2(3):273–292
Bosma W, Cannon J, Playoust C (1997) The Magma algebra system I: The user language. J Symbolic Comput 24(3–4):235–265
Carlet C, Guilley S (2018) Statistical properties of side-channel and fault injection attacks using coding theory. Cryptography Commun 10(5):909–933
Carlet C, Mesnager S, Tang C, Qi Y (2019) New characterization and parametrization of LCD codes. IEEE Trans Inform Theory 65(1):39–49
Caruso X, Le Borgne J (2017) A new faster algorithm for factoring skew polynomials over finite fields. J Symbolic Comput 79:411–443
Diao L, Gao J, Lu J (2020) Some results on \(\mathbb{Z}_{p}\mathbb{Z}_{p}[v]-\)additive cyclic codes. Adv Math Commun. https://doi.org/10.3934/amc.2020029
Dougherty ST, Kim J-L, Liu H (2010) Constructions of self-dual codes over finite commutative chain rings. Int J Inform Coding Theory 1(2):171–190
Ezerman MF, Ling S, Solé P, Yemen O (2011) From skew cyclic codes to asymetric quantum codes. Adv Math Com 5(1):44–57
Gao J, Ma F, Fu F (2017) Skew constacyclic codes over the ring \(\mathbb{F}_{q}+v\mathbb{F}_{q}\). Appl Comput Math 6(3):286–295
Gao J, Shen L, Fu F (2016) A Chinese remainder theorem approach to skew generalized quasi-cyclic codes. Cryptogr Commun 8(1):51–66
Grassl M, Gulliver TA (2009) On circulant self-dual codes over small fields. Des Codes Cryptogr 52(1):57–81
Giesbrecht M (1998) Factoring in skew-polynomial rings over finite fields. J Symbolic Comput 26(4):463–486
Guenda K, Gulliver TA (2015) Repeated root constacyclic codes of length \(mp^{s}\) over \(\mathbb{F}_{p^{m}}+u\mathbb{F} _{p^{m}}+\dots +u^{e-1}\mathbb{F}_{p^{m}}\), J. Algebra Applic., 14(1)
Jitman S, Ling S, Udomkavanich P (2012) Skew constacyclic codes over finite chain rings. Adv Math Commun 6(1):39–63
Jitman S, Sangwisut E, Udonkavanich P (2014) The Gray image of skew-constacyclic codes over \(\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}+\dots +u^{e-1}\mathbb{F}_{p^{m}}\). Chamchuri J Math 6:1–15
Qian J-F, Zhang L-N, Zhu S-X (2006) \((1+u)-\)cyclic codes over \(\mathbb{F}_{2}+u\mathbb{F}_{2}\). Appl Math Lett 19(8):820–823
Ling S, Solé P (2001) Type II codes over \(\mathbb{F}_{4}+u\mathbb{F}_{4}\). Europ J Combin 22(7):983–997
Liu X, Xu X (2014) Cyclic and negacyclic codes of length \(2p^{s}\) over \(\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}\). Acta Mathematica Scientia 34B(3):829–839
McDonald BR (1974) Finite Rings with Identity. Marcel Dekker, New York
Siap I, Abualrub T, Aydin N, Seneviratne P (2011) Skew cyclic codes of arbitrary length. Int J Inf Coding Theory 2(1):10–20
Sok L, Shi M (2018) P, Sol\(\acute{e}\), Constructions of optimal LCD codes over large finite fields. Finite Fields Their Appl 50:138–153
Wang Y, Gao J (2019) MacDonald codes over the ring \(\mathbb{F}_{p}+v\mathbb{F}_{p}+v^{2}\mathbb{F}_{p},\) Computational and Applied Mathematics, 38(4), 169
Wu M (2013) Skew cyclic and quasi-cyclic codes of arbitrary length over Galois rings. Int J Algebra 7(17):803–807
Yang X, Massey J-L (1993) Cyclic complementary dual codes. Discr Math 126:391–393
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The authors are greatly indebted to the editor and reviewers for their remarks and advice which allowed us to improve the paper considerably.
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Communicated by Eduardo Souza de Cursi.
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Hebbache, Z., Guenda, K., Özzaim, N.T. et al. Some skew constacyclic codes over \(\mathbb {F}_{q}+u\mathbb {F}_{q}\). Comp. Appl. Math. 40, 52 (2021). https://doi.org/10.1007/s40314-021-01425-6
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DOI: https://doi.org/10.1007/s40314-021-01425-6