Cyclic Codes over the Integers Modulopm

AbstractThe purpose of this paper is twofold. First, we generalize the results of Pless and Qian and those of Pless, Solé, and Qian for cyclic Z4-codes to cyclic Zpm-codes. Second, we establish connections between this new development and the results on cyclic Zpm-codes obtained by Calderbank and Sloane. We produce generators for the cyclic Zpm-codes which are analogs to those for cyclic Z4-codes. We show that these may be used to produce a single generator for such codes. In particular, this proves that the ringRn= Zpm[x]/(xn− 1) is principal, a result that had been previously announced with an incorrect proof. Generators for dual codes of cyclic Zpm-codes are produced from the generators of the corresponding cyclic Zpm-codes. In addition, we also obtain generators for the cyclicpm-ary codes induced from the idempotent generators for cyclicp-ary codes