An explicit expression for Euclidean self-dual cyclic codes of length 2^k over Galois ring GR(4,m)

07/19/2020 ∙ by Yuan Cao, et al. ∙ 0

For any positive integers m and k, existing literature only determines the number of all Euclidean self-dual cyclic codes of length 2^k over the Galois ring GR(4,m), such as in [Des. Codes Cryptogr. (2012) 63:105–112]. Using properties for Kronecker products of matrices of a specific type and column vectors of these matrices, we give a simple and efficient method to construct all these self-dual cyclic codes precisely. On this basis, we provide an explicit expression to accurately represent all distinct Euclidean self-dual cyclic codes of length 2^k over GR(4,m), using combination numbers. As an application, we list all distinct Euclidean self-dual cyclic codes over GR(4,m) of length 2^k explicitly, for k=4,5,6.

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