On rank and MDR cyclic and negacyclic codes of length [formula omitted] over [formula omitted

Abstract

In this paper, a set of generators (in a unique from) called the distinguished set of generators, of a cyclic code C of length n=2k (where k is a natural number) over Z2m is obtained. This set of generators is used to find the rank of the cyclic code C. It is proved that the rank of a cyclic code C of length n=2k over Z2m is equal to n−v, where v is the degree of a minimal degree polynomial in C. Then a description of all MHDR (maximum hamming distance with respect to rank) cyclic codes of length n=2k over Z2m is given. An example of the best cyclic codes over Z8 of length 4 having largest minimum Hamming, Lee and Euclidean distances among all cyclic codes of the same rank is also given. Further, using an isomorphism between cyclic and negacyclic codes of odd length over finite chain rings given by Dinh, Lopez and Szabo, the above results are extended to cyclic and negacyclic codes of length pk over Zpm for an odd prime p.