Free cyclic codes over finite local rings

Abstract

In [2] it was shown that a 1-generator quasi-cyclic code C of length n = ml of index l over ℤ4 is free if C is generated by a polynomial which divides Xm - 1. In this paper, we prove that a necessary and sufficient condition for a cyclic code over ℤpk of length m to be free is that it is generated by a polynomial which divides Xm - 1. We also show that this can be extended to finite local rings with a principal maximal ideal.