In $1998,$ Daemen {\it{ et al.}} introduced a circulant Maximum Distance Separable (MDS) matrix in the diffusion layer of the Rijndael block cipher, drawing significant attention to circulant MDS matrices. This block cipher is now universally acclaimed as the AES block cipher. In $2016,$ Liu and Sim introduced cyclic matrices by modifying the permutation of circulant matrices and established the existence of MDS property for orthogonal left-circulant matrices, a notable subclass within cyclic matrices. While circulant matrices have been well-studied in the literature, the properties of cyclic matrices are not. Back in $1961$, Friedman introduced $g$-circulant matrices which form a subclass of cyclic matrices. In this article, we first establish a permutation equivalence between a cyclic matrix and a circulant matrix. We explore properties of cyclic matrices similar to $g$-circulant matrices. Additionally, we determine the determinant of $g$-circulant matrices of order $2^d \times 2^d$ and prove that they cannot be simultaneously orthogonal and MDS over a finite field of characteristic $2$. Furthermore, we prove that this result holds for any cyclic matrix.

翻译：1998年，Daemen 等人将循环最大距离可分（MDS）矩阵引入 Rijndael 分组密码的扩散层，引发了学界对循环 MDS 矩阵的广泛关注。该分组密码现已被公认为 AES 分组密码。2016年，Liu 和 Sim 通过修改循环矩阵的置换方式引入了循环矩阵，并证明了循环矩阵中一个重要子类——正交左循环矩阵——的 MDS 性质存在性。尽管循环矩阵在文献中已得到充分研究，但循环矩阵的性质尚未被深入探讨。早在 1961 年，Friedman 引入了 $g$-循环矩阵，它是循环矩阵的一个子类。本文首先建立了循环矩阵与循环矩阵之间的置换等价关系，并探讨了循环矩阵与 $g$-循环矩阵类似的性质。此外，我们计算了 $2^d \times 2^d$ 阶 $g$-循环矩阵的行列式，并证明了在特征为 $2$ 的有限域上，这类矩阵不可能同时满足正交性与 MDS 性质。进一步地，我们证明了该结论对任意循环矩阵均成立。