On bases and the dimensions of twisted centralizer codes

10/05/2021

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by   Ahmad Muchlis, et al.

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Alahmadi et al. ["Twisted centralizer codes", Linear Algebra and its Applications 524 (2017) 235-249.] introduced the notion of twisted centralizer codes, 𝒞_𝔽_q(A,γ), defined as 𝒞_𝔽_q(A,γ)={ X ∈𝔽_q^n × n: AX=γ XA}, for A ∈𝔽_q^n × n, and γ∈𝔽_q. Moreover, Alahmadi et al. ["On the dimension of twisted centralizer codes", Finite Fields and Their Applications 48 (2017) 43-59.] 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.