Cyclic Codes for MDS and MHDR on Finite Chain Rings

Abstract

Coding theory aims to provide optimal codes for detecting and correcting a maximum number of errors during data
transmission through noisy channels. Cyclic codes have been in focus due to their rich algebraic structure which
enables easy encoding and decoding of data through the process of channel coding. Cyclic codes over rings have
gained a lot of importance after the remarkable break- through given by Hammons et al. in reference [1]. A vast
literature is available on the structure of cyclic codes over fields, integer residue rings, Galois rings, finite chain rings, and
some finite nonchain rings [2–29]. Cyclic codes over finite chain rings with length coprime to the characteristic of residue
field have been investigated in references [2, 16, 22]. Islam and Prakash have established a unique set of gener- ators for
cyclic codes over Zpk in reference [4] and for cycliccodes over Fq uFq, u2 0 in reference [5].