# On the parameters of intertwining codes

Research output: Contribution to journalArticle

### Abstract

Let F be a field and let Fr×s denote the space of r × s matrices over F. Given equinu-merous 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. © 2019 Society of Mathematicians Physicists and Astronomers of Slovenia. All rights reserved.
Original language English 49-58 10 Ars Mathematica Contemporanea 16 1 13 Sep 2018 https://doi.org/10.26493/1855-3974.1547.454 Published - 2019

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Polynomials
Minimum Distance
Linear Codes
Characteristic polynomial
Upper and Lower Bounds
Subspace
Denote
Generalise
Subset

### Cite this

@article{fa7826676f444a0abb088767b2849230,
title = "On the parameters of intertwining codes",
abstract = "Let F be a field and let Fr×s denote the space of r × s matrices over F. Given equinu-merous 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. {\circledC} 2019 Society of Mathematicians Physicists and Astronomers of Slovenia. All rights reserved.",
keywords = "Linear code, Dimension, Distance",
author = "Stephen Glasby and Cheryl Praeger",
year = "2019",
doi = "10.26493/1855-3974.1547.454",
language = "English",
volume = "16",
pages = "49--58",
journal = "Ars Mathematica Contemporanea",
issn = "1855-3974",
publisher = "DMFA Slovenije",
number = "1",

}

In: Ars Mathematica Contemporanea, Vol. 16, No. 1, 2019, p. 49-58.

Research output: Contribution to journalArticle

TY - JOUR

T1 - On the parameters of intertwining codes

AU - Glasby, Stephen

AU - Praeger, Cheryl

PY - 2019

Y1 - 2019

N2 - Let F be a field and let Fr×s denote the space of r × s matrices over F. Given equinu-merous 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. © 2019 Society of Mathematicians Physicists and Astronomers of Slovenia. All rights reserved.

AB - Let F be a field and let Fr×s denote the space of r × s matrices over F. Given equinu-merous 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. © 2019 Society of Mathematicians Physicists and Astronomers of Slovenia. All rights reserved.

KW - Linear code

KW - Dimension

KW - Distance

U2 - 10.26493/1855-3974.1547.454

DO - 10.26493/1855-3974.1547.454

M3 - Article

VL - 16

SP - 49

EP - 58

JO - Ars Mathematica Contemporanea

JF - Ars Mathematica Contemporanea

SN - 1855-3974

IS - 1

ER -