On the parameters of intertwining codes
S. P. Glasby, Cheryl E. Praeger · Ars Mathematica Contemporanea · 2018
Let F be a field and let Fr × s denote the space of r × s matrices over F. Given equinumerous subsets A = {Ai ∣ i ∈ I} ⊆ Fr × r and B = {Bi ∣ i ∈ I} ⊆ Fs × s we call the subspace C(A, B) := {X ∈ Fr × s ∣ AiX = XBi for i ∈ I} an intertwining code. We show that if C(A, B) ≠ {0}, then for each i ∈ I, the characteristic polynomials of Ai and Bi 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.