Determining Minimal Degree Polynomials of a Cyclic Code of Length $$2^k$$ over $$\mathbb {Z}_8$$

Abstract

The rank of a cyclic code of length \(n=2^k\) over \({\mathbb {Z}}_8\) is \(n-v\) where v is the degree of a minimal degree polynomial in the code. In this paper, minimal degree polynomials in a cyclic code C of length \(n = 2^k\) (where k is a natural number) over \({\mathbb {Z}}_8\) are determined. Further, using these minimal degree polynomials, all 95 (46 principally generated and 49 non principally generated) cyclic codes of length 4 over \({\mathbb {Z}}_8\) are calculated in terms of their distinguished sets of generators.