1-generator quasi-cyclic codes over finite chain rings

Abstract

Let R be an arbitrary commutative finite chain ring with \(1\ne 0\). 1-generator quasi-cyclic (QC) codes over R are considered in this paper. Let \(\gamma \) be a fixed generator of the maximal ideal of R, \(F=R/\langle \gamma \rangle \) and \(|F|=q\). For any positive integers m, n satisfying \(\mathrm{gcd}(q,n)=1\), let \(\mathcal{R}_n=R[x]/\langle x^n-1\rangle \). Then 1-generator QC codes over R of length mn and index m can be regarded as 1-generator \(\mathcal{R}_n\)-submodules of the module \(\mathcal{R}_n^m\). First, we consider the parity check polynomial of a 1-generator QC code and the properties of the code determined by the parity check polynomial. Then we give the enumeration of 1-generator QC codes with a fixed parity check polynomial in standard form over R. Finally, under the condition that \(\mathrm{gcd}(|q|_n,m)=1\), where \(|q|_n\) denotes the order of q modulo n, we describe an algorithm to list all distinct 1-generator quasi-cyclic codes with a fixed parity check polynomial in standard form over R of length mn and index m.