On the structure of $1$-generator quasi-polycyclic codes over finite chain rings

TL;DR

This paper characterizes the structure of 1-generator quasi-polycyclic codes over finite chain rings, providing generator and parity check polynomials, and constructs new quaternary codes with novel parameters using computational methods.

Contribution

It offers a detailed algebraic characterization of 1-generator QP codes over chain rings and introduces a method to determine their parity check polynomials.

Findings

Characterization of generator polynomials and minimal generating sets

Determination of parity check polynomials using Gröbner bases

Construction of new quaternary codes with improved parameters

Abstract

Quasi-polycyclic (QP for short) codes over a finite chain ring $R$ are a generalization of quasi-cyclic codes, and these codes can be viewed as an $R[x]$-submodule of $\mathcal{R}_m^{\ell}$, where $\mathcal{R}_m:= R[x]/\langle f\rangle$, and $f$ is a monic polynomial of degree $m$ over $R$. If $f$ factors uniquely into monic and coprime basic irreducibles, then their algebraic structure allow us to characterize the generator polynomials and the minimal generating sets of 1-generator QP codes as $R$-modules. In addition, we also determine the parity check polynomials for these codes by using the strong Gr\"{o}bner bases. In particular, via Magma system, some quaternary codes with new parameters are derived from these 1-generator QP codes.