Matrix-product structure of repeated-root constacyclic codes over finite fields

TL;DR

This paper demonstrates that certain repeated-root constacyclic codes over finite fields can be represented as matrix-product codes of nested simpler codes, revealing a new structural perspective.

Contribution

It establishes a novel matrix-product structure for repeated-root constacyclic codes over finite fields, generalizing previous results and providing a new approach for code analysis.

Findings

Any such constacyclic code is monomially equivalent to a matrix-product code.

The matrix-product structure involves a nested sequence of simpler constacyclic codes.

This structural insight aids in understanding and constructing these codes.

Abstract

For any prime number $p$, positive integers $m, k, n$ satisfying ${\rm gcd}(p,n)=1$ and $\lambda_0\in \mathbb{F}_{p^m}^\times$, we prove that any $\lambda_0^{p^k}$-constacyclic code of length $p^kn$ over the finite field $\mathbb{F}_{p^m}$ is monomially equivalent to a matrix-product code of a nested sequence of $p^k$ $\lambda_0$-constacyclic codes with length $n$ over $\mathbb{F}_{p^m}$.