On the parameters of intertwining codes

Stephen P. Glasby, Cheryl E. Praeger

Abstract


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.


Keywords


Linear code, dimension, distance

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DOI: https://doi.org/10.26493/1855-3974.1547.454

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications