Twisted Centralizer Codes

TL;DR

This paper introduces twisted centralizer codes, a new class of linear codes defined by matrix commutation relations, and explores their properties including dimensions, distances, and automorphisms.

Contribution

It defines twisted centralizer codes and analyzes their properties, revealing how the minimal distance varies with the scalar parameter, extending the understanding of matrix-based codes.

Findings

Minimal distance of centralizer codes is at most n.

For certain scalars, the minimal distance can reach n^2.

Properties like dimensions and automorphism groups are characterized.

Abstract

Given an $n\times n$ matrix $A$ over a field $F$ and a scalar $a\in F$, we consider the linear codes $C(A,a):=\{B\in F^{n\times n}\mid \,AB=aBA\}$ of length $n^2$. We call $C(A,a)$ a twisted centralizer code. We investigate properties of these codes including their dimensions, minimum distances, parity-check matrices, syndromes, and automorphism groups. The minimal distance of a centralizer code (when $a=1$) is at most $n$, however for $a\ne 0,1$ the minimal distance can be much larger, as large as $n^2$.