Characterization of cyclic codes over {ℬ[X;(1/m)Z0]}m > 1 and efficient encoding decoding algorithm for cyclic codes. Issue 5 (4th May 2017)

• Record Type: Journal Article
• Authors: Shah, Tariq; Azam, Naveed Ahmed
• Is Part Of: International journal of computer mathematics. Volume 94:Issue 5(2017)
• Page Start: 1015
• Page End: 1027
• Publication Date: 2017-05-04
• Subjects: Cyclic codes -- error detection and correction -- semigroup rings -- polynomial rings; 11T71 -- 20M25 -- 94B05 -- 94B40; Computers -- Periodicals; Numerical analysis -- Periodicals; Automation -- Periodicals; 004.0151
• DOI: 10.1080/00207160.2016.1158815
• Languages: English
• ISSNs: 0020-7160

Abstract

ABSTRACT: The aim of the paper is twofold. Firstly, cyclic codes of arbitrary length n over the family of semigroup rings{ B [ X ; ( 1 / m ) Z 0 ] } m > 1 are completely characterized in terms of ideals of the ringsB [ X ; ( 1 / m ) Z 0 ] / ( X n / m − 1 ), whereB is finite unitary commutative ring. Then, generator matrix for cyclic codes over{ B [ X ; ( 1 / m ) Z 0 ] } m > 1 is derived from their ideal representation. Secondly, an efficient encoding decoding algorithm is presented for polynomial cyclic codeC 0 of length n based on cyclic codeC over{ B [ X ; ( 1 / m ) Z 0 ] } m > 1 . Rigorous analyses are performed to examine the efficiency of proposed scheme. Analyses reveal that the proposed algorithm can simultaneously encode and decode m messages ofC 0 . Furthermore, the proposed technique is capable of detecting and correcting all burst errors up to length md − m and⌊ ( m d − m ) / 2 ⌋ from transmitted codewords ofC 0 . 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 ( n +1)n length binary cyclic code, An. Acad. Brasil. Ciênc. 85 (2013), pp. 863–872].