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

对于任何正整数m和k，现有文献仅确定长度为所有欧几里得自对偶循环码的数目$2^{ķ}$ 在伽罗瓦环上 $\mathrm{GR}（4，米）$，例如Kiah等。（2012）[17]。利用特定类型矩阵的Kronecker乘积的性质和这些矩阵的列向量，我们提供了一种简单有效的方法来精确构造所有这些自对偶循环码。在此基础上，我们提供了一个明确的表达式来准确表示所有长度不同的欧几里德自对偶循环码$2^{ķ}$ 超过 $\mathrm{GR}（4，米）$，使用二项式系数。作为应用程序，我们列出了所有不同的欧几里得自对偶循环码$\mathrm{GR}（4，米）$ 长度 $2^{ķ}$ 明确地，对于 $ķ=4，5，6$。