On MDS Property of g-Circulant Matrices

Abstract

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.