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 dimCF(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 dimCF(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 dimCF(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 dimCF(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 -