Basri ÇALIŞKAN, Kemal BALIKÇI

Skew Cyclic Codes over $\mathbb{Z}_{8}+u\mathbb{Z}_{8}+v\mathbb{Z}_{8}$

In this paper, we study the skew cyclic codes over the ring $S=\mathbb{Z}_{8}+u\mathbb{Z}_{8}+v\mathbb{Z}_{8}$, where $u^{2}=u$, $v^{2}=v$, $uv=vu=0$. We consider these codes as left $S[x,\theta]$-submodules and use the Gray map on $S$ to obtain the $\mathbb{Z}_{8}$-images. The generator and parity-check matrices of a free $\theta$-cyclic code of even length over $S$ are determined. Also, these codes are generalized to double skew-cyclic codes. We give some examples using Magma computational algebra system.

Keywords:

Double-cyclic codes, Gray map linear codes, skew-cyclic codes,

PDF

___

Bosma, W. Cannon J., Playoust, C., The Magma algebra system I. The user language, J. Symbolic Comput., 24(1997), 235–265.
Boucher, D., Ulmer, F., Coding with skew polynomial rings, J. of Symbolic Comput., 44(2009), 1644–1656.
Boucher, D., Geiselmann, W., Ulmer, F., Skew-cyclic codes, Appl. Alg. in Eng., Comm. and Comput., 18(4)(2007), 379–389.
Cengellenmis, Y., On the cyclic codes over F3 + vF3, Int. J. of Algebra, 4(6)(2010), 253–259.
Çalışkan, B., Balıkçı, K., Counting Z2Z4Z8-additive codes, European J. of Pure and Applied Math., 12(2)(2019), 668–679.
Çalışkan, B., Linear Codes over the Ring Z8 + uZ8 + vZ8, (ICOMAA-2020), Conference Proceeding Science and Technology, 3(1)(2020), 19–23.
Çalışkan, B., Cylic Codes over the Ring Z8 + uZ8 + vZ8, (ICMASE 2020), Proceedings Book, Ankara Hacı Bayram Veli University, Ankara, Turkey, (2020), 7–12.
Dertli, A., Cengellenmis, Y., On the codes over the ring Z4 + uZ4 + vZ4 cyclic, constacyclic, quasi-cyclic codes, their skew codes, cyclic DNA and skew cyclic DNA codes, Prespacetime Journal, 10(2)(2019), 196–213.
Gao, J., Skew cyclic codes over Fp + vFp, J. Appl. Math. Inform, 31(3-4)(2013), 337–342.
Hammons A.R., Kumar V., Calderbank A.R., Sloane N.J.A., Sole P., The Z4-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40(1994), 301–319.
Jin, L., Skew cyclic codes over ring Fp + vF2, J. of Electronics (China), 31(3)(2014), 228–231.
Melakhessou, A., Aydin, N., Hebbache, Z., Guenda, K., Zq(Zq + uZq)-linear skew constacyclic codes, J. Algebra Comb. Discrete Appl., 7(1)(2019), 85–101.
Mohammadi, R., Rahimi S., Mousavi, H., On skew cyclic codes over a finite ring, Iranian J. of Math. Sci. and Inf., 14(1)(2019), 135–145.
Sharma, A., Bhaintwal, M., A class of skew-constacyclic codes over Z4 + uZ4, Int. J. Inf. and Coding Theory, 4(4)(2017), 289–303.
Siap, I., Abualrub, T., Aydin, N., Seneviratne, P., Skew cyclic codes of arbitrary length, Int. J. of Inf. and Coding Theory, 2(1)(2011), 10–20.