Twisted centralizer codes

Given an n×n matrix A over a field F and a scalar a∈F, we consider the linear codes C(A,a):={B∈Fn×n|AB=aBA} of length n2. We call C(A,a) a twisted centralizer code. We investigate properties of these codes including their dimensions, minimum distances, parity-check matrices, syndromes, and automorphism groups. The minimal distance of a centralizer code (when a=1) is at most n, however for a≠0,1 the minimal distance can be much larger, as large as n2.