Circulant Maximum Distance Separable (MDS) matrices have gained significant importance due to their applications in the diffusion layer of the AES block cipher. In $2013$, Gupta and Ray established that circulant involutory matrices of order greater than $3$ cannot be MDS over $\mathbb{F}_{2^m}$. This finding prompted a generalization of circulant matrices and the involutory property of matrices by various authors. In $2016$, Liu and Sim introduced cyclic matrices by changing the permutation of circulant matrices. In $1961,$ Friedman introduced $g$-circulant matrices which form a subclass of cyclic matrices. In this article, we first discuss $g$-circulant matrices with involutory and MDS properties. We prove that $g$-circulant involutory matrices of order $k \times k$ cannot be MDS unless $g \equiv -1 \pmod k.$ Next, we delve into $g$-circulant semi-involutory and semi-orthogonal matrices with entries from finite fields. We establish that the $k$-th power of the associated diagonal matrices of a $g$-circulant semi-orthogonal (semi-involutory) matrix of order $k \times k$ results in a scalar matrix. These findings extend the recent results on circulant matrices established by Kumar {\it{et al.}} $(2026)$ and Chatterjee {\it{et al.}} $(2022)$. Furthermore, we prove that cyclic matrices of order $2^{d} \times 2^{d}$ over finite fields of characteristic $2$ cannot simultaneously possess both the MDS and semi-orthogonal properties.

翻译：循环最大距离可分（MDS）矩阵因在AES分组密码扩散层中的应用而具有重要意义。2013年，Gupta和Ray证明，在$\mathbb{F}_{2^m}$上，阶数大于3的循环对合矩阵不可能是MDS的。这一发现促使多位研究者推广了循环矩阵及矩阵的对合性质。2016年，Liu和Sim通过改变循环矩阵的置换引入了循环矩阵。1961年，Friedman引入了g-循环矩阵，它是循环矩阵的一个子类。本文首先讨论了具有对合和MDS性质的g-循环矩阵。我们证明了，除非满足$g \equiv -1 \pmod k$，否则$k \times k$阶g-循环对合矩阵不可能是MDS的。接着，我们深入研究了元素取自有限域的g-循环半对合矩阵和半正交矩阵。我们证明了$k \times k$阶g-循环半正交（半对合）矩阵的相伴对角矩阵的$k$次幂是一个标量矩阵。这些结果推广了Kumar等人（2026年）和Chatterjee等人（2022年）近期关于循环矩阵的研究成果。此外，我们证明了在特征为2的有限域上，$2^{d} \times 2^{d}$阶循环矩阵不能同时具有MDS和半正交性质。