Let $s,m$ be the positive integers and $p$ be any prime number. Next, let $GR(p^s,p^{sm})$ be a Galois ring of characteristic $p^s$ and cardinality $p^{sm}$. In the present paper, we explore the construction of Cauchy MDS matrices over Galois rings. Moreover, we introduce a new approach that considers nilpotent elements and Teichmüller set of Galois ring $GR(p^s,p^{sm})$ to reduce the number of entries in these matrices. Furthermore, we construct $p^{(s-1)m}(p^m-1)$ distinct functions with the help of Frobenius automorphisms. These functions preserve MDS property of matrices. Finally, we prove some results using automorphisms and isomorphisms of the Galois rings that can be used to generate new Cauchy MDS matrices.

翻译：设 $s,m$ 为正整数，$p$ 为任意素数。记 $GR(p^s,p^{sm})$ 为特征 $p^s$、基数 $p^{sm}$ 的Galois环。本文研究了Galois环上Cauchy MDS矩阵的构造方法。进一步地，我们提出一种新思路，利用Galois环 $GR(p^s,p^{sm})$ 的幂零元与Teichmüller集来减少这类矩阵中的元素数量。此外，借助Frobenius自同构，我们构造了 $p^{(s-1)m}(p^m-1)$ 个不同的函数，这些函数能保持矩阵的MDS性质。最后，我们利用Galois环的自同构与同构性质证明了一系列结果，这些结果可用于生成新的Cauchy MDS矩阵。