Alahmadi, Adel; Alamoudi, Shefa; Karadeniz, Suat; Yildiz, Bahattin; Praeger, Cheryl; Solé, Patrick Centraliser codes. (English) Zbl 1298.94127 Linear Algebra Appl. 463, 68-77 (2014). Summary: Centraliser codes are codes of length \(n^2\) defined as centralisers of a given matrix \(A\) of order \(n\). Their dimension, parity-check matrices, syndromes, and automorphism groups are investigated. A lower bound on the dimension is \(n\), the order of \(A\). This bound is met when the minimal polynomial is equal to the annihilator, i.e. for so-called cyclic (a.k.a. non-derogatory) matrices. If, furthermore, the matrix is separable and the adjacency matrix of a graph, the automorphism group of that graph is shown to be abelian and to be even trivial if the alphabet field is of even characteristic. Cited in 5 Documents MSC: 94B05 Linear codes (general theory) 13M05 Structure of finite commutative rings Keywords:group centralisers; matrix codes; cyclic matrices; separable matrices PDFBibTeX XMLCite \textit{A. Alahmadi} et al., Linear Algebra Appl. 463, 68--77 (2014; Zbl 1298.94127) Full Text: DOI References: [1] Akbari, S.; Ghandehari, M.; Hadian, M.; Mohammadian, A., On commuting graphs of semisimple rings, Linear Algebra and its Appl., 390, 345-355 (2004) · Zbl 1063.05087 [2] Biggs, N., Algebraic Graph Theory (1996), Cambridge University Press: Cambridge University Press Cambridge [3] Broxson, B. J., The Kronecker product (2006), University of North Florida, PhD dissertation [4] Chao, C.-Y., A note on the eigenvalues of a graph, J. Combin. Theory Ser. B, 10, 301-302 (1971) · Zbl 0213.51002 [5] Criscuolo, G.; Kwok, C.-M.; Mowshowitz, A.; Tortora, R., The group and the minimal polynomial of a graph, J. Combin. Theory Ser. B, 29, 293-302 (1980) · Zbl 0452.05050 [6] Grassl, M., Online tables of linear codes [7] Mowshowitz, A., The group of a graph whose adjacency matrix has all distinct eigenvalues, (Proof Techniques in Graph Theory, Proc. Second Ann Arbor Graph Theory Conf.. Proof Techniques in Graph Theory, Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, MI, 1968 (1969), Academic Press: Academic Press New York), 109-110 [8] Neumann, P. M.; Praeger, C. E., Cyclic matrices over finite fields, J. of the London Math Soc. (2), 52, 263-284 (1995) · Zbl 0839.15011 [9] Williams, V. V., An overview of the recent progress on matrix multiplication (2012), ACM SIGACT News This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.