2024/06/22 by Tapas Chatterjee, Chatterjee, Tapas, Ayantika Laha +1
Computer Science · Engineering · Physics and Astronomy · #Matrix Theory and Algorithms #graph theory and CDMA systems #Advanced Mathematical Theories and Applications
paper · pdf · doi:10.48550/arxiv.2406.15872
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 \mathbbF2m. 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 \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 × k results in a scalar matrix. These findings extend the recent results on circulant matrices established by Kumar \itet al. (2026) and Chatterjee \itet al. (2022). Furthermore, we prove that cyclic matrices of order 2d × 2d over finite fields of characteristic 2 cannot simultaneously possess both the MDS and semi-orthogonal properties.