On Rank and MDR Cyclic Codes of Length $$2^k$$ 2 k Over $$Z_8$$ Z 8

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 \(n = 2^k\) (where k is a natural number) over \(Z_8\) 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 \(n=2^k\) over \(Z_8\) 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 \(n=2^k\) over \(Z_8\) is given. An example of the best codes over \(Z_8\) of length 4 having largest minimum Hamming, Lee and Euclidean distances among all codes of the same rank is also given.