Galois Correspondence on Linear Codes over Finite Chain Rings

TL;DR

This paper explores Galois correspondence for linear codes over finite chain rings, establishing invariance criteria, and providing bounds and applications to cyclic codes, enhancing understanding of code structure over ring extensions.

Contribution

It introduces Galois operators for codes over finite chain rings, characterizes Galois invariance via generator matrix entries, and applies the theory to bounds and cyclic code structures.

Findings

Galois invariance characterized by generator matrix entries in fixed ring

Established Galois correspondence for S-linear codes

Provided bounds and applications to cyclic codes

Abstract

Given $\texttt{S}|\texttt{R}$ a finite Galois extension of finite chain rings and $\mathcal{B}$ an $\texttt{S}$-linear code we define two Galois operators, the closure operator and the interior operator. We proof that a linear code is Galois invariant if and only if the row standard form of its generator matrix has all entries in the fixed ring by the Galois group and show a Galois correspondence in the class of $\texttt{S}$-linear codes. As applications some improvements of upper and lower bounds for the rank of the restriction and trace code are given and some applications to $\texttt{S}$-linear cyclic codes are shown.