Abstract
Given a field F, a scalar lambda is an element of F and a matrix A is an element of F-nxn, the twisted centralizer code C-F(A, lambda) := {B is an element of F-nxn vertical bar AB - lambda BA = 0} is a linear code of length n(2) over F. When A is cyclic and A 0 we prove that dim C-F(A, lambda) = deg(gcd(c(A) (t), lambda(n)c(A) (lambda(-1)t))) where c(A) (t) denotes the characteristic polynomial of A. We also show how C-F(A, lambda) decomposes, and we estimate the probability that C-F(A, lambda) is nonzero when vertical bar F vertical bar is finite. Finally, we prove dim C-F(A, lambda) <= n(2)/2 for lambda is not an element of{0,1} and 'almost all' n x n matrices A over F. Crown Copyright (C) 2017 Published by Elsevier Inc. All rights reserved.