On the Counting of Involutory MDS Matrices

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 $\mathbb{F}_{2^r}$.