On the parameters of intertwining codes - Glasby, Stephen P. (Ars mathematica contemporanea, 2019)
Let ?$F$? be a field and let ?$F^{r × s}$? denote the space of ?$r × s$? matrices over ?$F$?. Given equinumerous subsets ?$\mathcal A = \{A_i \mid i \in I\} \subseteq F^{r × r}$? and ?$\mathcal B = \{B_i \mid i \in I\} \subseteq F^{s × s}$? we call the subspace ?$C(\mathcal{A, B}) := \{X \in F^{r × s} \mid A_iX = XB_i\ for\ i \in I\}$? an intertwining code. We show that if ?$C(\mathcal{A, B}) ≠ \{0\}$?, 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 ?$k = \dim(C(\mathcal{A, B}))$? and give upper and lower bounds. This generalizes previous work. 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 ?$C(\mathcal{A, B})$? are denoted by ?$d, k,\ and\ R = k/rs$?, respectively.
Keywords: dimension; dimenzija; distance; linear code; linearna koda; razdalja