On bases and the dimensions of twisted centralizer codes

TL;DR

This paper generalizes previous work on twisted centralizer codes by providing exact dimension formulas and an explicit basis construction algorithm for any matrix, extending beyond cyclic or diagonalizable cases.

Contribution

It determines the exact dimension of twisted centralizer codes for all matrices and offers an explicit basis construction algorithm, improving upon prior bounds.

Findings

Exact dimension formulas for all matrices.

Algorithm for explicit basis construction.

Extension beyond cyclic or diagonalizable matrices.

Abstract

Alahmadi et al. ["Twisted centralizer codes", \emph{Linear Algebra and its Applications} {\bf 524} (2017) 235-249.] introduced the notion of twisted centralizer codes, $\mathcal{C}_{\mathbb{F}_q}(A,\gamma),$ defined as \[ \mathcal{C}_{\mathbb{F}_q}(A,\gamma)=\lbrace X \in \mathbb{F}_q^{n \times n}:~\ AX=\gamma XA\rbrace, \] for $A \in \mathbb{F}_q^{n \times n},$ and $\gamma \in \mathbb{F}_q.$ Moreover, Alahmadi et al. ["On the dimension of twisted centralizer codes", \emph{Finite Fields and Their Applications} {\bf 48} (2017) 43-59.] also investigated the dimension of such codes and obtained upper and lower bounds for the dimension, and the exact value of the dimension only for cyclic or diagonalizable matrices $A.$ Generalizing and sharpening Alahmadi et al.'s results, in this paper, we determine the exact value of the dimension as well as provide an algorithm to construct an explicit basis of the codes for any given matrix $A.$