Characterization of cyclic codes over {[bernou][X; (1/m)Z sub(0)]} sub(m > 1) and efficient encoding decoding algorithm for cyclic codes

• Published in: International journal of computer mathematics Vol. 94; no. 5; pp. 1015 - 1027
• Main Authors: Shah, Tariq, Azam, Naveed Ahmed
• Format: Journal Article
• Language: English
• Published: 04.05.2017
• Subjects: Algorithms; Coding; Decoding; Encoding; Image transmission; Mathematical analysis; Rings (mathematics)
• ISSN: 0020-7160; 1029-0265
• DOI: 10.1080/00207160.2016.1158815

The aim of the paper is twofold. Firstly, cyclic codes of arbitrary length n over the family of semigroup rings [Image omitted.] are completely characterized in terms of ideals of the rings [Image omitted.], where [Image omitted.] is finite unitary commutative ring. Then, generator matrix for cyclic codes over [Image omitted.] is derived from their ideal representation. Secondly, an efficient encoding decoding algorithm is presented for polynomial cyclic code [Image omitted.] of length n based on cyclic code [Image omitted.] over [Image omitted.]. Rigorous analyses are performed to examine the efficiency of proposed scheme. Analyses reveal that the proposed algorithm can simultaneously encode and decode m messages of [Image omitted.]. Furthermore, the proposed technique is capable of detecting and correcting all burst errors up to length md-m and [Image omitted.] from transmitted codewords of [Image omitted.]. The newly developed algorithm is also compared with some of the existing coding techniques. It is evident from comparison that the proposed scheme is better than existing coding techniques in Babu and Zimmermann [Decoding of linear codes over galois rings, IEEE Trans. Inform. Theory, 47 (2001)], Nagpaul and Jain [Topics in Applied Abstract Algebra, The Brooks/Cole Series in Advanced Mathematics, 2005], Shah et al. [A method for improving the code rate and error correction capability of a cyclic code, Comput. Appl. Math. 32 (2013), pp. 261-274, doi:10.1007/s40314-013-0010-1] and Shah et al. [A decoding method of an n length binary BCH code through (n1)n length binary cyclic code An. Acad. Brasil. Cienc. 85 (2013), pp. 863-872].