Cyclic codes of length 5ps over Fpm + uFpm and their duals

Author:

Dinh Hai1,Nguyen Bac2,Tansuchat Roengchai3,Thi Hiep4

Affiliation:

1. Department of Mathematical Sciences, Kent State University, Ohio, USA

2. Institute of Fundamental and Applied Sciences, Duy Tan University, Ho Chi Minh City, Vietnam + Faculty of Natural Sciences, Duy Tan University, Da Nang, Vietnam

3. Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, Thailand

4. Faculty of Education, Thu Dau Mot University, Binh Duong, Vietnam

Abstract

For an odd prime p ? 5, the structures of cyclic codes of length 5ps over R = Fpm + uFpm (u2 = 0) are completely determined. Cyclic codes of length 5ps over R are considered in 3 cases, namely, p ? 1 (mod 5), p ? 4 (mod 5), p ? 2 or 3 (mod 5). When p ? 1 (mod 5), a cyclic code of length 5ps over R can be expressed as a direct sum of a cyclic code and ?ps i -constacyclic codes of length ps over R, where ?ps i = ?i(pm?1)ps/10, i = 1,3,7,9. When p ? 4 (mod 5), it is equivalent to pm ? 1 (mod 5) when m is even and pm ? 4 (mod 5) when m is odd. If pm ? 1 (mod 5) when m is even, then a cyclic code of length 5ps over R can be obtained as a direct sum of a cyclic code and ?ps i -constacyclic codes of length ps over R, where ?psi = ?i(pm?1)ps/10, i = 1,3,7,9. If pm ?/ 4 (mod 5) when m is odd, then a cyclic code of length 5ps over R can be expressed as a direct sum of a cyclic code of length ps over R and an ?1 and ?2-constacyclic code of length 2ps over R, for some ?1, ?2 ? Fpm\{0}. If p ? 2 or 3 (mod 5) such that pm ?/ 1 (mod 5), then a cyclic code of length 5ps over R can be expressed as C1 ? C2, where C1 is an ideal of R[x]/?xps?1? and C2 is an ideal of R[x]/(x4+x3+x2+x+1)ps ?. We also investigate all ideals of R[x]/?(x4+x3+x2+x+1)ps ? to study detail structure of a cyclic code of length 5ps over R. In addition, dual codes of all cyclic codes of length 5ps over R are also given. Furthermore, we give the number of codewords in each of those cyclic codes of length 5ps over R. As cyclic and negacyclic codes of length 5ps over R are in a one-by-one equivalent via the ring isomorphism x ? ?x, all our results for cyclic codes hold true accordingly to negacyclic codes.

Publisher

National Library of Serbia

Reference54 articles.

1. T. Abualrub and R. Oehmke, On the generators of Z4 cyclic codes of length 2e, IEEE Trans. Inform. Theory 49 (2003), 2126-2133.

2. M.M. Al-Ashker, Simplex codes over the ring F2 + uF2, Arab. J. Sci. Eng. Sect. A Sci. 30 (2005), 277-285.

3. E. Bannai, M. Harada, T. Ibukiyama, A. Munemasa, and M. Oura, Type II codes over F2 + uF2 and applications to Hermitian modular forms, Abh. Math. Sem. Univ. Hamburg 73 (2003), 13-42.

4. E.R. Berlekamp, Algebraic Coding Theory, revised 1984 edition, Aegean Park Press, 1984.

5. E. R. Berlekamp, Negacyclic codes for the Lee metric, in: Proceedings of the Conference on Combinatorial Mathematics and Its Application, Chapel Hill, NC, 1968, 298-316.

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