  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
$2022.$
