• Mousavi, Hamed (Department of Mathematics Tarbiat Modares University) ;
  • Moussavi, Ahmad (Department of Mathematics Tarbiat Modares University) ;
  • Rahimi, Saeed (Department of Information Technology Emam Hossein University)
  • Received : 2015.06.25
  • Accepted : 2018.02.01
  • Published : 2018.11.30


In this paper, we study an special type of cyclic codes called skew cyclic codes over the ring ${\mathbb{F}}_p+v{\mathbb{F}}_p+v^2{\mathbb{F}}_p$, where p is a prime number. This set of codes are the result of module (or ring) structure of the skew polynomial ring (${\mathbb{F}}_p+v{\mathbb{F}}_p+v^2{\mathbb{F}}_p$)[$x;{\theta}$] where $v^3=1$ and ${\theta}$ is an ${\mathbb{F}}_p$-automorphism such that ${\theta}(v)=v^2$. We show that when n is even, these codes are either principal or generated by two elements. The generator and parity check matrix are proposed. Some examples of linear codes with optimum Hamming distance are also provided.



  1. T. Abualrub, A. Ghrayeb, N. Aydin, and I. Siap, On the construction of skew quasicyclic codes, IEEE Trans. Inform. Theory 56 (2010), no. 5, 2081-2090.
  2. T. Blackford, Negacyclic codes over $Z_4$ of even length, IEEE Trans. Inform. Theory 49 (2003), no. 6, 1417-1424.
  3. A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over $F_2$ + $uF_2$, IEEE Trans. Inform. Theory 45 (1999), no. 4, 1250-1255.
  4. D. Boucher, W. Geiselmann, and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engrg. Comm. Comput. 18 (2007), no. 4, 379-389.
  5. D. Boucher, P. Sole, and F. Ulmer, Skew constacyclic codes over Galois rings, Adv. Math. Commun. 2 (2008), no. 3, 273-292.
  6. D. Boucher and F. Ulmer, A note on the dual codes of module skew codes, in Cryptography and coding, 230-243, Lecture Notes in Comput. Sci., 7089, Springer, Heidelberg, 2011.
  7. A. R. Calderbank and N. J. A. Sloane, Modular and p-adic cyclic codes, Des. Codes Cryptogr. 6 (1995), no. 1, 21-35.
  8. P.-L. Cayrel, C. Chabot, and A. Necer, Quasi-cyclic codes as codes over rings of matrices, Finite Fields Appl. 16 (2010), no. 2, 100-115.
  9. R. Dastbasteh, H. Mousavi, A. Abualrub, N. Aydin, and J. Haghighat, Skew cyclic codes over ${\mathbb{F}}_p$ + $u{\mathbb{F}}_p$, International J. Information and Coding Theory, Accepted, 2018.
  10. S. T. Dougherty and Y. H. Park, On modular cyclic codes, Finite Fields Appl. 13 (2007), no. 1, 31-57.
  11. J. Gao, Skew cyclic codes over $F_p$ + $vF_p$, J. Appl. Math. Inform. 31 (2013), no. 3-4, 337-342.
  12. L. Jin, Skew cyclic codes over ring $F_p$ + $vF_p$, J. Electronics (China) 31 (2014), no. 3, 228-231.
  13. D. Mandelbaum, An application of cyclic coding to message identification, IEEE Transactions on Communication Technology 17 (1969), no. 1, 42-48.
  14. P. Kanwar and S. R. Lopez-Permouth, Cyclic codes over the integers modulo $p^m$, Finite Fields Appl. 3 (1997), no. 4, 334-352.
  15. V. S. Pless and Z. Qian, Cyclic codes and quadratic residue codes over $Z_4$, IEEE Trans. Inform. Theory 42 (1996), no. 5, 1594-1600.
  16. I. Siap, T. Abualrub, N. Aydin, and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. J. Inf. Coding Theory 2 (2011), no. 1, 10-20.
  17. K. Tokiwa, M. Kasahara, and T. Namekawa, Burst-error-correction capability of cyclic codes, Electron. Comm. Japan 66 (1983), no. 11, 60-66.
  18. J. Wolfmann, Binary images of cyclic codes over ${\mathbb{Z}}_4$, IEEE Trans. Inform. Theory 47 (2001), no. 5, 1773-1779.