On the parameters of intertwining codes

Abstract

Let F be a field and let F^r × s denote the space of r × s matrices over F. Given equinumerous subsets A = {A_i ∣ i ∈ I} ⊆ F^r × r and B = {B_i ∣ i ∈ I} ⊆ F^s × s we call the subspace C(A, B) := {X ∈ F^r × s ∣ A_iX = XB_i for i ∈ I} an intertwining code. We show that if C(A, B) ≠ {0}, then for each i ∈ I, the characteristic polynomials of A_i and B_i and share a nontrivial factor. We give an exact formula for k = dim(C(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(A, B) are denoted by d, k, and R = k/rs, respectively.