A Class of LCD Codes Through Cyclic Codes Over $ZpR$

Abstract

In recent time, some mixed types of alphabets have been considered for constructing error correcting codes. These constructions include $\bbbz_{2}\bbbz_{4}-$additive codes, $\bbbz_{2}\bbbz_{2}[u]-$linear codes et cetera. In this paper, we studied a class of codes over a mixed ring $\bbbz_{p}R$ where $R=\bbbz_{p}+v\bbbz_{p}+v^{2}\bbbz_{p}, v^{3}=v.$ We determined an algebraic structure of these codes under certain conditions. We have also constructed a class of LCD cyclic codes over $\bbbz_{p}R$. A necessary and sufficient condition for a cyclic code to be a complementary dual (LCD) code has been obtained.

Keywords

• Linear Cyclic Codes
• Codes Over Mixed Alphabets
• Gray Map
• LCD Codes

References

1. P. Delsarte. An algebraic approach to the association schemes of coding theory. PhD thesis, Universite Catholique de Louvain, 1973.
2. T. Abualrub, I. Siap, and N. Aydin. $ZZ_{2}$ $ZZ_{4}$ additive cyclic codes. IEEE Trans. Inf. Theory, 3:1508-1514, 2014.
3. J. Borges, C. Fernandez-Cordoba, and R. Ten-Valls. $ZZ_{2}$ $ZZ_{4}$ additive cyclic codes, generator polynomials and dual codes. IEEE Trans. Inf. Theory, 11:6348-6354, 2016.
4. I. Aydogdu and T. Abualrub. The structure of $ZZ_{2}$ $ZZ_{2^{8}}$additive cyclic codes. Discrete Math. Algorithms Appl., 4:1850048-1850060, 2018.
5. I. Aydogdu and I. Siap. On $ZZ_{p^{r}}$ $ZZ_{p^{s}}$additive codes. Linear Multilinear Algebra, 10:2089-2102, 2014.
6. I. Aydogdu and T. Abualrub. The structure of $ZZ_{2}$ $ZZ_{2}[u]$ cyclic and constacyclic codes. IEEE Trans. Inf. Theory, 63(8):4883-4893, 2017.
7. L. Diao and J. Gao. $ZZ_{p}$ $ZZ_{p}[u]$additive cyclic codes. Int. J. Inf. Coding Theory, 1:1-17, 2018.
8. B. Srinivasulu and B. Maheshanand. $ZZ_{2}$($ZZ_{2}$ + $uZZ_{2}$) additive cyclic codes and their duals. Discrete Math. Algorithms Appl., 2:1650027-1650045, 2016.

9. Z. Hebbache, A. Kaya, N. Aydin, and K. Guenda. On some skew codes over $ZZ_{q}$ + $uZZ_{q}$. Discrete Mathematics Algorithms and Applications, 2022.
10. J-L. Massey. Linear codes with complementary duals. Discrete Math., 106-107:337-342, 1992.
11. C. Carlet. Boolean Functions for Cryptography and Error Correcting Codes. Cambridge University Press, Cambridge, U.K., 2010.
12. X. Liu and H. Liu. Lcd codes over finite chain rings. Finite Fields Appl., 34:1-19, 2015.
13. C. Li, C. Ding, and S. Li. Lcd cyclic codes over finite fields. IEEE Trans. Inf. Theory, 63:4344-4356, 2017.
14. X. Yang and J-L. Massey. The condition for a cyclic code to have a complementary dual. Discrete Math., 126:391-393, 1994.
15. L. Diao, J. Gao, and J. Lu. Some results on $ZZ_{p}$$ZZ_{p}[v]$ additive cyclic codes. Adv. Math. Commun., 4:555-572, 2020.
16. M. Bhaintwal and S-K. Wasan. On quasi-cyclic codes over $ZZ_{p}$:. Appl. Algebra Engrg. Comm. Comput., 20:459-480, 2009.