On the parameters of intertwining codes

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}$={B_i\mid i\in I}\subseteq F^{s\times s}$wecallthesubspace$C($\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 $C(\mathcal{A},\mathcal{B})\mathrm{\ne }\{0\}$, 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 $d=rs\mathrm{/}k=1\mathrm{/}R$ is large where the minimum distance, dimension, and rate of the linear code $C(\mathcal{A},\mathcal{B})$ by $d$, $k$, and $R=k\mathrm{/}rs$, respectively.