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Centraliser codes. (English) Zbl 1298.94127

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.

MSC:

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