Matrix-product structure of constacyclic codes over finite chain rings $\mathbb{F}_{p^m}[u]/\langle u^e\rangle$

TL;DR

This paper characterizes $(1+ ext{omega} u)$-constacyclic codes over a finite chain ring as matrix-product codes of cyclic codes, providing a new structural understanding and an iterative construction method for these codes.

Contribution

It establishes a matrix-product structure for constacyclic codes over finite chain rings and introduces an iterative construction method based on shorter-length codes.

Findings

Constacyclic codes are monomially equivalent to matrix-product codes.

Provides a construction method for codes of arbitrary length from shorter codes.

Offers a new perspective on the structure of constacyclic codes over finite chain rings.

Abstract

Let $m,e$ be positive integers, $p$ a prime number, $\mathbb{F}_{p^m}$ be a finite field of $p^m$ elements and $R=\mathbb{F}_{p^m}[u]/\langle u^e\rangle$ which is a finite chain ring. For any $\omega\in R^\times$ and positive integers $k, n$ satisfying ${\rm gcd}(p,n)=1$, we prove that any $(1+\omega u)$-constacyclic code of length $p^kn$ over $R$ is monomially equivalent to a matrix-product code of a nested sequence of $p^k$ cyclic codes with length $n$ over $R$ and a $p^k\times p^k$ matrix $A_{p^k}$ over $\mathbb{F}_p$. Using the matrix-product structures, we give an iterative construction of every $(1+\omega u)$-constacyclic code by $(1+\omega u)$-constacyclic codes of shorter lengths over $R$.