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 - λ+ \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 - λ$ factors as $ x^m - λ= \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^{°f_i} - 1 \right), $ where $r$ is the multiplicative order of $λ$, and \( N \) is the number of integers \( k \) such that $ 0 \leq k < \left\lfloor \frac{m - 1}{r} \right\rfloor + 1 \quad \text{and} \quad \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.

翻译：在有限域$\mathbb{F}_q$上，若矩阵$M$的所有方子矩阵均为非奇异矩阵，则称其为最大距离可分（MDS）矩阵。这类MDS矩阵在密码学与编码理论中至关重要，因其能提供强大的数据保护并有效促进信息扩散。本文引入一种新型矩阵——常数-$g$-循环矩阵，该矩阵扩展了$g$-循环矩阵的概念。这类矩阵源于由多项式$h(x) = x^m - λ+ \sum_{i=0}^{m-1} h_i x^i$在$\mathbb{F}_q$上定义的线性变换。我们确定了此类矩阵存在的上界，并给出了判定其可逆性的条件，这有助于判断其是否为MDS矩阵。若多项式$x^m - λ$可分解为$x^m - λ= \prod_{i=1}^{t} f_i(x)^{e_i}$，其中每个$f_i(x)$为不可约多项式，则可逆常数-$g$-循环矩阵的数量为$N \cdot \prod_{i=1}^{t} \left( q^{°f_i} - 1 \right)$。此处$r$为$λ$的乘法阶，$N$为满足$0 \leq k < \left\lfloor \frac{m - 1}{r} \right\rfloor + 1$且$\gcd(1 + rk, m) = 1$的整数$k$的数量。该公式有助于缩减判定此类矩阵是否为MDS的验证范围。此外，我们完整刻画了3阶与4阶$g$-循环MDS矩阵的特性。受斜多项式环启发，我们还构建了一种新型$g$-循环矩阵变体。最后，我们提供了若干与研究成果相关的示例。