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On bases and the dimensions of twisted centralizer codes. (English) Zbl 1467.94054

Summary: A. Alahmadi et al. [Linear Algebra Appl. 524, 235–249 (2017; Zbl 1380.94141)] introduced the notion of twisted centralizer codes, \(\mathcal{C}_{\mathbb{F}_q}(A,\gamma)\), defined as \[ \mathcal{C}_{\mathbb{F}_q}(A,\gamma)=\{X\in\mathbb{F}_q^{n\times n}:AX=\gamma XA\}, \] for \(A\in\mathbb{F}_q^{n\times n}\), and \(\gamma\in\mathbb{F}_q\). Moreover, A. Alahmadi et al. [Finite Fields Appl. 48, 43–59 (2017; Zbl 1398.94243)] 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 [loc. cit.], 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 \(A\).

MSC:

94B05 Linear codes (general theory)
13M05 Structure of finite commutative rings
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References:

[1] Alahmadi, A.; Alamoudi, S.; Karadeniz, S.; Yildiz, B.; Praeger, C. E.; Solé, P., Centraliser codes, Linear Algebra Appl., 463, 68-77 (2014) · Zbl 1298.94127
[2] Alahmadi, A.; Glasby, S. P.; Praeger, C. E.; Solé, P.; Yildiz, B., Twisted centralizer codes, Linear Algebra Appl., 524, 235-249 (2017) · Zbl 1380.94141
[3] Alahmadi, A.; Glasby, S. P.; Praeger, C. E., On the dimension of twisted centralizer codes, Finite Fields Appl., 48, 43-59 (2017) · Zbl 1398.94243
[4] Huffman, W. C.; Pless, V., Fundamentals of Error Correcting Codes (2003), Cambridge University Press · Zbl 1099.94030
[5] Joydep, P.; Maurya, P. K.; Mukherjee, S.; Bagchi, S., Generalized twisted centralizers codes (2017)
[6] Roman, S., Advanced Linear Algebra (2007), Springer: Springer New York
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