On MDS Property of g-Circulant Matrices

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 × k cannot be MDS unless g ≡ -1 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 × k results in a scalar matrix. These findings can be viewed as an extension of the results concerning circulant matrices established by Chatterjee {et al.} in $2022.$