Cyclic codes over galois rings

Abstract

Let R be a Galois ring of characteristic pa, where p is a prime and a is a natural number. In this paper cyclic codes of arbitrary length n over R have been studied. The generators for such codes in terms of minimal degree polynomials of certain subsets of codes have been obtained.We prove that a cyclic code of arbitrary length n over R is generated by at most min{a, t+1} elements, where t = max{deg(g(x))}, g(x) a generator. In particular, it follows that a cyclic code of arbitrary length n over finite fields is generated by a single element. Moreover, the explicit set of generators so obtained turns out to be a minimal strong Gröbner basis.