A note on MDS Property of Circulant Matrices

• URL: http://arxiv.org/abs/2406.16973v1
• Date: Sat, 22 Jun 2024 16:00:00 GMT
• Title: A note on MDS Property of Circulant Matrices
• Authors: Tapas Chatterjee, Ayantika Laha,
• Abstract summary: In $2014$, Gupta and Ray proved that the circulant involutory matrices over the finite field $mathbbF_2m$ can not be maximum distance separable (MDS) This article delves into circulant matrices possessing these characteristics over the finite field $mathbbF_2m$.
• Score: 3.069335774032178
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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