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

Year 2023, , 96 - 103, 30.06.2023

Basri Çalışkan , Kemal Balıkçı

https://doi.org/10.47000/tjmcs.995569

Abstract

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

References

• 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.