On the parameters of intertwining codes

TL;DR

This paper studies intertwining codes formed by matrices over a field, providing formulas for their dimensions, bounds, and constructions with large minimum distance, generalizing previous results in the area.

Contribution

It derives an exact formula for the dimension of intertwining codes, establishes bounds, and constructs codes with large minimum distance, extending prior work in the field.

Findings

Dimension formula for intertwining codes

Bounds on code parameters

Examples of codes with large minimum distance

Abstract

Let $F$ be a field and let $F^{r\times s}$ denote the space of $r\times s$ matrices over $F$. Given equinumerous subsets $\mathcal{A}=\{A_i\mid i \in I\}\subseteq F^{r\times r}$ and $\mathcal{B}=\{B_i\mid i\in I\}\subseteq F^{s\times s}$ we call the subspace $C(\mathcal{A},\mathcal{B}):=\{X\in F^{r\times s}\mid A_iX=XB_i\ {\rm for }\ i\in I\}$ an \emph{intertwining code}. We show that if $C(\mathcal{A},\mathcal{B})\ne\{0\}$, then for each $i\in I$, the characteristic polynomials of $A_i$ and $B_i$ and share a nontrivial factor. We give an exact formula for $k=\dim(C(\mathcal{A},\mathcal{B}))$ and give upper and lower bounds. This generalizes previous work in this area. Finally we construct intertwining codes with large minimum distance when the field is not `too small'. We give examples of codes where $d=rs/k=1/R$ is large where the minimum distance, dimension, and rate of the linear code $C(\mathcal{A},\mathcal{B})$ by $d$, $k$, and $R=k/rs$, respectively.