In $2014$, Gupta and Ray proved that the circulant involutory matrices over the finite field $\mathbb{F}_{2^m}$ can not be maximum distance separable (MDS). This non-existence also extends to circulant orthogonal matrices of order $2^d \times 2^d$ over finite fields of characteristic $2$. These findings inspired many authors to generalize the circulant property for constructing lightweight MDS matrices with practical applications in mind. Recently, in $2022,$ Chatterjee and Laha initiated a study of circulant matrices by considering semi-involutory and semi-orthogonal properties. Expanding on their work, this article delves into circulant matrices possessing these characteristics over the finite field $\mathbb{F}_{2^m}.$ Notably, we establish a correlation between the trace of associated diagonal matrices and the MDS property of the matrix. We prove that this correlation holds true for even order semi-orthogonal matrices and semi-involutory matrices of all orders. Additionally, we provide examples that for circulant, semi-orthogonal matrices of odd orders over a finite field with characteristic $2$, the trace of associated diagonal matrices may possess non-zero values.

翻译：2014年，Gupta与Ray证明了有限域$\mathbb{F}_{2^m}$上的循环对合矩阵不可能具有最大距离可分（MDS）性质。该不存在性结论可进一步推广至特征为2的有限域上$2^d \times 2^d$阶循环正交矩阵。这些发现促使众多学者推广循环性质以构造具有实际应用的轻量级MDS矩阵。近期于2022年，Chatterjee与Laha通过考虑半对合与半正交性质开启了循环矩阵的新研究方向。本文在其工作基础上，深入探究有限域$\mathbb{F}_{2^m}$上具有此类特性的循环矩阵。特别地，我们建立了关联对角矩阵的迹与矩阵MDS性质之间的对应关系。我们证明该对应关系对偶数阶半正交矩阵及任意阶半对合矩阵均成立。此外，我们通过示例说明：在特征为2的有限域上，对于奇数阶循环半正交矩阵，其关联对角矩阵的迹可能具有非零值。