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.