The optimal branch number of MDS matrices has established their prominence in the design of diffusion layers for various block ciphers and hash functions. Consequently, several matrix structures have been proposed for designing MDS matrices, including Hadamard and circulant matrices. In this paper, we first provide the count of Hadamard MDS matrices of order $4$ over the field $\mathbb{F}_{2^r}$. Subsequently, we present the counts of order $2$ MDS matrices and order $2$ involutory MDS matrices over the field $\mathbb{F}_{2^r}$. Finally, leveraging these counts of order $2$ matrices, we derive an upper bound for the number of all involutory MDS matrices of order $4$ over $\mathbb{F}_{2^r}$.

翻译：MDS矩阵的最优分支数使其在多种分组密码和哈希函数的扩散层设计中占据重要地位。为此，学界已提出多种矩阵结构用于构造MDS矩阵，包括Hadamard矩阵和循环矩阵。本文首先给出域$\mathbb{F}_{2^r}$上$4$阶Hadamard MDS矩阵的计数结果。随后，我们提出了域$\mathbb{F}_{2^r}$上$2$阶MDS矩阵和$2$阶对合MDS矩阵的计数方法。最后，基于这些$2$阶矩阵的计数结果，推导出$\mathbb{F}_{2^r}$上所有$4$阶对合MDS矩阵数量的上界。