On the generalization of $g$-circulant MDS matrices

Abstract

A matrix $M$ over the finite field ${\mathbb{F}}_q$ is called \emph{maximum distance separable} (MDS) if all of its square submatrices are non-singular. These MDS matrices are very important in cryptography and coding theory because they provide strong data protection and help spread information efficiently. In this paper, we introduce a new type of matrix called a \emph{consta-$g$-circulant matrix}, which extends the idea of $g$-circulant matrices. These matrices come from a linear transformation defined by the polynomial $h(x)=x^m-\hat{I}»+{\sum }_{i=0}^{m-1}{h}_i{x}^i$ over ${\mathbb{F}}_q$. We find the upper bound of such matrices exist and give conditions to check when they are invertible. This helps us know when they are MDS matrices. If the polynomial $x^m-\hat{I}»$ factors as $x^m-\hat{I}»={\prod }_{i=1}^t{f}_i(x{)}^{e_i},$ where each \( f_i(x) \) is irreducible, then the number of invertible consta-$g$-circulant matrices is $N\cdot {\prod }_{i=1}^t\left(q^{\hat{A}°f_i}-1\right),$ where $r$ is the multiplicative order of $\hat{I}»$, and \( N \) is the number of integers \( k \) such that $0\le k<\lfloor \frac{m-1}{r}\rfloor +1\text{and}\gcd (1+rk,m)=1.$ This formula help us to reduce the number of cases to check whether such matrices is MDS. Moreover, we give complete characterization of $g$-circulant MDS matrices of order 3 and 4. Additionally, inspired by skew polynomial rings, we construct a new variant of $g$-circulant matrix. In the last, we provide some examples related to our findings.