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

Abstract

For any positive integers m and k, existing literature only determines the number of all Euclidean self-dual cyclic codes of length 2k over the Galois ring GR(4,m), such as in Kiah et al. (2012) [17]. 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 represent accurately all distinct Euclidean self-dual cyclic codes of length 2k over GR(4,m), using binomial coefficients. As an application, we list all distinct Euclidean self-dual cyclic codes over GR(4,m) of length 2k explicitly, for k=4,5,6.