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.
Categories: math.CO