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设计扩散层的首选。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。虽然穷举搜索适用于小阶MDS矩阵，但由于搜索空间巨大，对于较大阶数的矩阵更倾向于直接构造法。文献中，已有大量关于使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，在安全性和效率之间提供了比MDS矩阵更好的平衡。然而，文献中尚无直接构造递归NMDS矩阵的方法。本文介绍了非递归和递归设置下NMDS矩阵的一些直接构造方法。此外，还提出了基于广义范德蒙矩阵的非递归MDS矩阵的直接构造方法。我们提出了一种使用广义范德蒙矩阵构造对合MDS和NMDS矩阵的方法。最后，我们证明了文献中与NMDS码相关的一些经验结论。