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 new 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, it derives the count of Hadamard Near-MDS (NMDS) and involutory Hadamard NMDS matrices, each with exactly one zero in each row, of order $4$ over $\mathbb{F}_{2^r}$. Furthermore, the paper discusses some circulant-like matrices for constructing NMDS matrices and proves that when $n$ is even, any $2n \times 2n$ Type-II circulant-like matrix can never be an NMDS matrix. While it is known that NMDS matrices may be singular, this paper establishes that singular Hadamard matrices can never be NMDS matrices. Moreover, it proves that there exist exactly two orthogonal Type-I circulant-like 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种分组密码和哈希函数扩散层中的重要性。因此，包括Hadamard矩阵和循环矩阵在内的多种矩阵结构被提出用于构造MDS矩阵。此外，在文献中，通常重点关注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，本文采取了一种不同的方法，不强调效率问题或引入新的构造。相反，其主要目标是枚举域$\mathbb{F}_{2^r}$上$4$阶Hadamard MDS矩阵和对合Hadamard MDS矩阵。具体而言，本文给出了域$\mathbb{F}_{2^r}$上$4$阶Hadamard MDS矩阵和对合Hadamard MDS矩阵计数的显式公式。此外，还推导了域$\mathbb{F}_{2^r}$上每行恰好有一个零元的$4$阶Hadamard近MDS（NMDS）矩阵和对合Hadamard NMDS矩阵的计数。进一步地，本文讨论了一些用于构造NMDS矩阵的类循环矩阵，并证明了当$n$为偶数时，任何$2n \times 2n$型II类循环矩阵都不可能是NMDS矩阵。虽然已知NMDS矩阵可能是奇异的，但本文证明了奇异Hadamard矩阵永远不可能是NMDS矩阵。此外，还证明了在域$\mathbb{F}_{2^r}$上恰好存在两个$4$阶正交型I类循环矩阵。