Let p be an odd prime and m and s positive integers, with m even. Let further \({\mathbb {F}}_{p^m}\) be the finite field of \(p^m\) elements and \(R={\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}\) (\(u^2=0\)). Then R is a finite chain ring of \(p^{2m}\) elements, and there is a Gray map from \(R^N\) onto \({\mathbb {F}}_{p^m}^{2N}\) which preserves distance and orthogonality, for any positive integer N. It is an interesting approach to obtain self-dual codes of length 2N over \({\mathbb {F}}_{p^m}\) by constructing self-dual codes of length N over R. In particular, it has been shown that one of the key problems in constructing self-dual repeated-root cyclic codes over R is to find an effective way to present precisely Hermitian self-dual cyclic codes of length \(p^s\) over R. But so far, only the number of these codes has been determined in literature. In this paper, we give an efficient way of constructing all distinct Hermitian self-dual cyclic codes of length \(p^s\) over R by using column vectors of Kronecker products of matrices with specific types. Furthermore, we provide an explicit expression to present precisely all these Hermitian self-dual cyclic codes, using binomial coefficients.

设p为奇素数，m和s为正整数，m为偶数。进一步令\({\mathbb {F}}_{p^m}\)为\(p^m\)元素的有限域且\(R={\mathbb {F}}_{p^m} +u{\mathbb {F}}_{p^m}\) ( \(u^2=0\) )。则R是\(p^{2m}\)个元素的有限链环，并且存在从\(R^N\)到\({\mathbb {F}}_{p^m}^ {2N}\)保留距离和正交性，对于任何正整数N。获得长度为 2 N的自对偶码是一种有趣的方法\({\mathbb {F}}_{p^m}\)通过在R上构造长度为N的自对偶码。特别是，已经表明，在R上构造自对偶重复根循环码的关键问题之一是找到一种有效的方法来精确呈现长度为\(p^s\)的Hermitian 自对偶循环码。 R。 _ 但到目前为止，文献中只确定了这些代码的数量。在本文中，我们给出了一种在 R 上构造长度为 \(p^s\) 的所有不同 Hermitian 自对偶循环码的有效方法通过使用具有特定类型的矩阵的克罗内克积的列向量。此外，我们使用二项式系数提供了一个明确的表达式来精确地呈现所有这些 Hermitian 自对偶循环码。