### Abstract

Original language | English |
---|---|

Pages (from-to) | 49-58 |

Number of pages | 10 |

Journal | Ars Mathematica Contemporanea |

Volume | 16 |

Issue number | 1 |

Early online date | 13 Sep 2018 |

DOIs | |

Publication status | Published - 2019 |

### Fingerprint

### Cite this

}

*Ars Mathematica Contemporanea*, vol. 16, no. 1, pp. 49-58. https://doi.org/10.26493/1855-3974.1547.454

**On the parameters of intertwining codes.** / Glasby, Stephen; Praeger, Cheryl.

Research output: Contribution to journal › Article

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 -