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

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 ${\rm GR}(4,m)$, 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 ${\rm GR}(4,m)$, using combination numbers. As an application, we list all distinct Euclidean self-dual cyclic codes over ${\rm GR}(4,m)$ of length $2^k$ explicitly, for $k=4,5,6$.