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Skew cyclic codes over \(\mathbb{F}_4R\). (English) Zbl 1496.94091

The paper under review deals with skew cyclic codes over a new alphabet set which is the ring \(\mathbb{F}_4R\), where \(\mathbb{F}_4\) is the field of \(4\) elements and \(R=\{ a+vb\mid a,b\in \mathbb{F}_4\} \) is the commutative ring with 16 elements, where \(v^2 = v\). The authors first study the algebraic properties of such codes and their duals. For instance, they show that skew cyclic codes over \(\mathbb{F}_4R\) are left \(R[X, \theta]\)-submodules of \(R_{\alpha, \beta} :=\mathbb{F}_4[X]/<X^\alpha-1>\times R[X, \theta]/<X^\beta-1>\). Then, they derive conditions for skew cyclic codes over \(\mathbb{F}_4R\) to be self-orthogonal. Using the Gray mapping, it is shown that the Gray image of any skew cyclic code over \(\mathbb{F}_4R\) is the product of a cyclic code over \(\mathbb{F}_4\) of length \(\alpha\) and two skew cyclic codes, each of length \(\beta\) over \(\mathbb{F}_4\). As an application, they construct optimal linear codes over \(\mathbb{F}_4\) as images of skew cyclic codes over \(\mathbb{F}_4R\) under the Gray mapping.

MSC:

94B15 Cyclic codes
94B05 Linear codes (general theory)

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References:

[1] Abualrub, T. and Aydin, N., Additive cyclic codes over mixed alphabets and the football pool problem, Discrete Math. Algorithms Appl.9(1) (2017) 1750010. · Zbl 1357.94088
[2] Abualrub, T., Ghrayeb, A., Aydin, N. and Siap, I., On the construction of skew quasi-cyclic codes, IEEE Trans. Inf. Theory56 (2010) 2081-2090. · Zbl 1366.94632
[3] Abualrub, T., Ghrayeb, A. and Zeng, X. N., Construction of cyclic codes over \(GF(4)\) for DNA computing, J. Franklin Inst.343 (2006) 448-457. · Zbl 1151.94013
[4] Abualrub, T., Siap, I. and Aydin, N., \( \Bbb Z_2 \Bbb Z_4\)-additive cyclic codes, IEEE Trans. Inf. Theory60 (2014) 1508-1514. · Zbl 1360.94398
[5] Bayram, A., Oztas, E. and Siap, I., Codes over \(\mathbb{F}_4+v \mathbb{F}_4\) and some DNA applications, Des. Codes Cryptogr.80 (2016) 379-393. · Zbl 1402.94078
[6] Borges, J., Fernández-Córdoba, C., Pujol, J., Rifà, J. and Villanueva, M., \( \Bbb Z_2 \Bbb Z_4\)-linear codes: Generator matrices and duality, Des. Codes Cryptogr.54 (2010) 167-179. · Zbl 1185.94092
[7] Boucher, D., Geiselmann, W. and Ulmer, F., Skew-cyclic codes, Appl. Algebra Eng. Commun. Comput.18 (2007) 379-389. · Zbl 1159.94390
[8] Boucher, D., Sole, P. and Ulmer, F., Skew constacyclic codes over Galois rings, Adv. Math. Commun.2 (2008) 273-292. · Zbl 1207.94085
[9] Z. Chen, Online database of quasi-twisted codes, http://www.tec.hkr.se/\( \sim\) chen/research/codes/searchqc2.htm, accessed on April 26, 2019.
[10] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, http://www.codetables.de, accessed on May 12, 2019.
[11] Gursoy, F., Siap, I. and Yildiz, B., Construction of skew cyclic codes over \(\mathbb{F}_q+v \mathbb{F}_q\), Adv. Math. Commun.8 (2014) 313-322. · Zbl 1300.94121
[12] D. Limbachiya, B. Rao and M. K. Gupta, The art of DNA strings: Sixteen years of DNA coding theory, Online at http://arXiv.org/pdf/1607.00266.
[13] Marathe, A., Condon, A. E. and Corn, R. M., On combinatorial DNA word design, J. Comput. Biol.8 (2001) 201-219. · Zbl 0969.68070
[14] Rifá-Pous, H., Rifá, J. and Ronquillo, L., \( \Bbb Z_2 \Bbb Z_4\)-additive perfect codes in steganography, Adv. Math. Commun.5 (2011) 425-433. · Zbl 1228.68027
[15] Siap, I., Abualrub, T., Aydin, N. and Seneviratne, P., Skew cyclic codes of arbitrary length, Int. J. Inf. Coding Theory2 (2011) 425-433. · Zbl 1320.94103
[16] Zhu, S., Wang, Y. and Shi, M., Some result on cyclic codes over \(\mathbb{F}_2+v \mathbb{F}_2\), IEEE Trans. Inf. Theory56 (2010) 1680-1684. · Zbl 1366.94651
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