On The Existance of MDS Matrices over \(\mathbb{F}_{p}+v\mathbb{F}_{p}\)
Defita, Intan Muchtadi-Alamsyah

Mathematics, Bandung Institute of Technology
Jalan Ganesha 10, Bandung 40132, Indonesia

Abstract

An \(n\times n\) matrix is called MDS (Maximum Distance Separable) matrix if and only if its submatrices are non-singular. In 2022, Adhiguna et al proved that over a field of characteristic \(p>2\) there is no orthogonal circulant MDS matrix of even order \(m\) and of order divisible by \(p\). In this research, we observe the existence of MDS matrices over ring \(\mathbb{F}_{p}+v\mathbb{F}_{p}\) where \(v^{2}=v\). Using the fact that for every \(a+vb\in \mathbb{F}_{p}+v\mathbb{F}_{p}\) can be written as \(a+vb=v(a+b)\oplus (1-v)a\), we prove that there is no orthogonal circulant MDS matrix of even order and of order divisible by \(p\) over \(\mathbb{F}_{p}+v\mathbb{F}_{p}\).

Keywords: MDS matrix- orthogonal- circulant

Topic: MATHEMATICS AND STATISTICS