On bases and the dimensions of twisted centralizer codes

Abstract Alahmadi et al. (2017) [2] introduced the notion of twisted centralizer codes, C F q ( A , γ ) , defined as C F q ( A , γ ) = { X ∈ F q n × n : A X = γ X A } , for A ∈ F q n × n , and γ ∈ F q . Moreover, Alahmadi et al. (2017) [3] also investigated the dimension of such codes and obtained upper and lower bounds for the dimension, and the exact value of the dimension only for cyclic or diagonalizable matrices A. Generalizing and sharpening Alahmadi et al.'s results, in this paper, we determine the exact value of the dimension as well as provide an algorithm to construct an explicit basis of the codes for any given matrix A.