On the study of semi-involutory and semi-orthogonal matrices

Abstract

This paper investigates semi-involutory and semi-orthogonal matrices, presenting two algorithms for verifying these properties for $n\times n$ matrices over $F_{p^m}^{⁎}$. The algorithms significantly reduce computational complexity by avoiding the need for non-singular diagonal matrices. The structure of circulant matrices with these properties is also explored, including the derivation of the exact form of their corresponding diagonal matrices. We further investigate MDS matrices with semi-involutory and semi-orthogonal properties. We prove that semi-involutory circulant matrices of order $n\ge 3$ over $F_{2^m}$, where $\gcd (n,2^m-1)=1$, cannot be MDS matrices. Additionally, we show that semi-orthogonal circulant matrices of order $2^n$ over $F_{2^m}$ cannot be MDS. Finally, explicit formulas for counting $3\times 3$ semi-involutory and semi-orthogonal MDS matrices over $F_{2^m}$ are derived, and exact counts for $4\times 4$ semi-involutory and semi-orthogonal MDS matrices over $F_{2^m}$ for $m=3$ and 4 are provided.