On the dimension of twisted centralizer codes

Adel Alahmadi; S. P. Glasby; Cheryl E. Praeger

doi:10.1016/j.ffa.2017.07.005

On the dimension of twisted centralizer codes

Adel Alahmadi, S. P. Glasby, Cheryl E. Praeger

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

104 Downloads (Pure)

Abstract

Given a field F, a scalar λ∈F and a matrix A∈F^n×n, the twisted centralizer code C_F(A,λ):={B∈F^n×n|AB−λBA=0} is a linear code of length n² over F. When A is cyclic and λ≠0 we prove that dim⁡C_F(A,λ)=deg⁡(gcd⁡(c_A(t),λⁿc_A(λ⁻¹t))) where c_A(t) denotes the characteristic polynomial of A. We also show how C_F(A,λ) decomposes, and we estimate the probability that C_F(A,λ) is nonzero when |F| is finite. Finally, we prove dim⁡C_F(A,λ)⩽n²/2 for λ∉{0,1} and ‘almost all’ n×n matrices A over F.

Original language	English
Pages (from-to)	43-59
Number of pages	17
Journal	Finite Fields and Their Applications
Volume	48
DOIs	https://doi.org/10.1016/j.ffa.2017.07.005
Publication status	Published - 1 Nov 2017

Access to Document

10.1016/j.ffa.2017.07.005

Alahmadi et al 2017 Dimension of twisted centralizer codesLicence: CC BY-NC-ND

Cite this

@article{573b399c991047478d4acc6278cd516f,

title = "On the dimension of twisted centralizer codes",

abstract = "Given a field F, a scalar λ∈F and a matrix A∈Fn×n, the twisted centralizer code CF(A,λ):={B∈Fn×n|AB−λBA=0} is a linear code of length n2 over F. When A is cyclic and λ≠0 we prove that dim⁡CF(A,λ)=deg⁡(gcd⁡(cA(t),λncA(λ−1t))) where cA(t) denotes the characteristic polynomial of A. We also show how CF(A,λ) decomposes, and we estimate the probability that CF(A,λ) is nonzero when |F| is finite. Finally, we prove dim⁡CF(A,λ)⩽n2/2 for λ∉{0,1} and {\textquoteleft}almost all{\textquoteright} n×n matrices A over F.",

keywords = "Dimension, Linear code, Twisted centralizer code",

author = "Adel Alahmadi and Glasby, {S. P.} and Praeger, {Cheryl E.}",

year = "2017",

month = nov,

day = "1",

doi = "10.1016/j.ffa.2017.07.005",

language = "English",

volume = "48",

pages = "43--59",

journal = "Finite Fields and Their Applications",

issn = "1071-5797",

publisher = "Academic Press",

}

TY - JOUR

T1 - On the dimension of twisted centralizer codes

AU - Alahmadi, Adel

AU - Glasby, S. P.

AU - Praeger, Cheryl E.

PY - 2017/11/1

Y1 - 2017/11/1

N2 - Given a field F, a scalar λ∈F and a matrix A∈Fn×n, the twisted centralizer code CF(A,λ):={B∈Fn×n|AB−λBA=0} is a linear code of length n2 over F. When A is cyclic and λ≠0 we prove that dim⁡CF(A,λ)=deg⁡(gcd⁡(cA(t),λncA(λ−1t))) where cA(t) denotes the characteristic polynomial of A. We also show how CF(A,λ) decomposes, and we estimate the probability that CF(A,λ) is nonzero when |F| is finite. Finally, we prove dim⁡CF(A,λ)⩽n2/2 for λ∉{0,1} and ‘almost all’ n×n matrices A over F.

AB - Given a field F, a scalar λ∈F and a matrix A∈Fn×n, the twisted centralizer code CF(A,λ):={B∈Fn×n|AB−λBA=0} is a linear code of length n2 over F. When A is cyclic and λ≠0 we prove that dim⁡CF(A,λ)=deg⁡(gcd⁡(cA(t),λncA(λ−1t))) where cA(t) denotes the characteristic polynomial of A. We also show how CF(A,λ) decomposes, and we estimate the probability that CF(A,λ) is nonzero when |F| is finite. Finally, we prove dim⁡CF(A,λ)⩽n2/2 for λ∉{0,1} and ‘almost all’ n×n matrices A over F.

KW - Dimension

KW - Linear code

KW - Twisted centralizer code

UR - http://www.scopus.com/inward/record.url?scp=85026727086&partnerID=8YFLogxK

U2 - 10.1016/j.ffa.2017.07.005

DO - 10.1016/j.ffa.2017.07.005

M3 - Article

AN - SCOPUS:85026727086

SN - 1071-5797

VL - 48

SP - 43

EP - 59

JO - Finite Fields and Their Applications

JF - Finite Fields and Their Applications

ER -

On the dimension of twisted centralizer codes

Abstract

Access to Document

Other files and links

Fingerprint

Cite this