The optimal branch number of MDS matrices has established their importance in designing diffusion layers for various block ciphers and hash functions. As a result, numerous matrix structures, including Hadamard and circulant matrices, have been proposed for constructing MDS matrices. Also, in the literature, significant attention is typically given to identifying MDS candidates with optimal implementations or proposing new constructions across different orders. However, this paper takes a different approach by not emphasizing efficiency issues or introducing novel constructions. Instead, its primary objective is to enumerate Hadamard MDS and involutory Hadamard MDS matrices of order $4$ within the field $\mathbb{F}_{2^r}$. Specifically, it provides an explicit formula for the count of both Hadamard MDS and involutory Hadamard MDS matrices of order $4$ over $\mathbb{F}_{2^r}$. Additionally, the paper presents the counts of order $2$ MDS matrices and order $2$ involutory MDS matrices over $\mathbb{F}_{2^r}$. Finally, leveraging these counts of order $2$ matrices, an upper bound is derived for the number of all involutory MDS matrices of order $4$ over $\mathbb{F}_{2^r}$.

翻译：MDS矩阵的最优分支数确立了其在设计各类分组密码和哈希函数扩散层中的重要性。因此，众多矩阵结构（包括Hadamard矩阵和循环矩阵）被提出用于构造MDS矩阵。此外，在现有文献中，研究重点通常集中于识别具有最优实现的MDS候选矩阵，或提出不同阶数的新构造方法。然而，本文采取了不同的研究路径，不强调效率问题或引入新颖构造，其主要目标是枚举域$\mathbb{F}_{2^r}$上$4$阶Hadamard MDS矩阵及对合Hadamard MDS矩阵。具体而言，本文给出了$\mathbb{F}_{2^r}$上$4$阶Hadamard MDS矩阵与$4$阶对合Hadamard MDS矩阵数量的显式计算公式。此外，论文还给出了$\mathbb{F}_{2^r}$上$2$阶MDS矩阵与$2$阶对合MDS矩阵的计数结果。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。