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. 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 can be viewed as an extension of the results concerning circulant matrices established by Chatterjee {\it{et al.}} in $2022.$

翻译：循环最大距离可分矩阵因其在AES分组密码扩散层中的应用而备受关注。2013年，Gupta和Ray证明了阶数大于3的循环对合矩阵不可能具有MDS性质。这一发现促使多位学者对循环矩阵及其对合性质进行推广。2016年，Liu和Sim通过改变循环矩阵的置换方式引入了循环矩阵。1961年，Friedman提出的g-循环矩阵构成了循环矩阵的一个子类。本文首先探讨具有对合性与MDS性质的g-循环矩阵。我们证明k×k阶g-循环对合矩阵不可能具有MDS性质，除非满足g ≡ -1 (mod k)。接着，我们深入研究有限域上元素的g-循环半对合与半正交矩阵。我们证明：k×k阶g-循环半正交（半对合）矩阵的伴随对角矩阵的k次幂可化为标量矩阵。这些结论可视为对Chatterjee等人在2022年建立的循环矩阵相关结果的扩展。