2016/07/20 by S. P. Glasby, Cheryl E. Praeger, Glasby, S. P. +3
Computer Science · Mathematics · #60C05 #94B65 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #cs.IT #math.AC #math.CO #math.IT #msc:60C05 #msc:94B65
paper · pdf · doi:10.48550/arxiv.1607.05838
17 pages, 2 figures Proof of Theorem 2.8 altered: last line and third last line changed
arxiv created 2017/07/14 · arxiv updated 2017/07/17
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. When A is cyclic and λ≠0 we prove that dim CF(A,λ)=deg(gcd(cA(t),λn cA(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)\leqslant n2/2 for λ\not∈\0,1\ and `almost all' matrices A.