The optimal branch number of MDS matrices makes them a preferred choice for designing diffusion layers in many block ciphers and hash functions. Consequently, various methods have been proposed for designing MDS matrices, including search and direct methods. While exhaustive search is suitable for small order MDS matrices, direct constructions are preferred for larger orders due to the vast search space involved. In the literature, there has been extensive research on the direct construction of MDS matrices using both recursive and nonrecursive methods. On the other hand, in lightweight cryptography, Near-MDS (NMDS) matrices with sub-optimal branch numbers offer a better balance between security and efficiency as a diffusion layer compared to MDS matrices. However, no direct construction method is available in the literature for constructing recursive NMDS matrices. This paper introduces some direct constructions of NMDS matrices in both nonrecursive and recursive settings. Additionally, it presents some direct constructions of nonrecursive MDS matrices from the generalized Vandermonde matrices. We propose a method for constructing involutory MDS and NMDS matrices using generalized Vandermonde matrices. Furthermore, we prove some folklore results that are used in the literature related to the NMDS code.

翻译：MDS矩阵的最优分支数使其成为许多分组密码和哈希函数中扩散层设计的首选。因此，人们提出了多种设计MDS矩阵的方法，包括搜索法和直接构造法。穷举搜索适用于小阶MDS矩阵，而由于大阶矩阵的搜索空间巨大，直接构造法更受青睐。文献中已有大量关于采用递归和非递归方法直接构造MDS矩阵的研究。另一方面，在轻量级密码学中，具有次优分支数的近MDS（NMDS）矩阵作为扩散层，相比MDS矩阵能更好地平衡安全性与效率。然而，目前文献中尚无直接构造递归NMDS矩阵的方法。本文介绍了非递归与递归情形下NMDS矩阵的一些直接构造方法，同时给出了基于广义范德蒙德矩阵的非递归MDS矩阵的直接构造。我们提出了一种利用广义范德蒙德矩阵构造对合MDS与NMDS矩阵的方法。此外，我们还证明了文献中与NMDS码相关的一些未公开结论。