On the dimension of twisted centralizer codes

TL;DR

This paper investigates the structure and dimension of twisted centralizer codes over finite fields, providing explicit formulas, decomposition methods, probability estimates, and bounds for their dimensions based on matrix properties.

Contribution

It introduces a formula for the dimension of twisted centralizer codes when matrices are cyclic and extends understanding of their structure and bounds.

Findings

Dimension formula involving gcd of characteristic polynomials

Decomposition of twisted centralizer codes

Upper bound of n^2/2 for most matrices when λ not in {0,1}

Abstract

Given a field $F$, a scalar $\lambda\in F$ and a matrix $A\in F^{n\times n}$, the twisted centralizer code $C_F(A,\lambda):=\{B\in F^{n\times n}\mid AB-\lambda BA=0\}$ is a linear code of length $n^2$. When $A$ is cyclic and $\lambda\ne0$ we prove that $\dim C_F(A,\lambda)=\mathrm{deg}(\gcd(c_A(t),\lambda^n c_A(\lambda^{-1}t)))$ where $c_A(t)$ denotes the characteristic polynomial of $A$. We also show how $C_F(A,\lambda)$ decomposes, and we estimate the probability that $C_F(A,\lambda)$ is nonzero when $|F|$ is finite. Finally, we prove $\dim C_F(A,\lambda)\leqslant n^2/2$ for $\lambda\not\in\{0,1\}$ and `almost all' matrices $A$.