Maximum Distance Separable (MDS) matrices play a central role in coding theory and symmetric-key cryptography due to their optimal diffusion properties. In this paper, we present a construction of MDS matrices using skew polynomial rings \( \mathbb{F}_q[X;θ,δ] \), where \( θ\) is an automorphism and \( δ\) is a \( θ\)-derivation on \( \mathbb{F}_q \). We introduce the notion of \( δ_θ \)-circulant matrices and study their structural properties. Necessary and sufficient conditions are derived under which these matrices are involutory and satisfy the MDS property. The resulting $δ_θ$-circulant matrix can be viewed as a generalization of classical constructions obtained in the absence of $θ$-derivations. One of the main contribution of this work is the construction of quasi recursive MDS matrices. In the setting of the skew polynomial ring $\mathbb{F}_q[X;θ]$, we construct quasi recursive MDS matrices associated with companion matrices. These matrices are shown to be involutory, yielding a strict improvement over the quasi-involutory constructions previously reported in the literature. Several illustrative results and examples are also provided.

翻译：最大距离可分（MDS）矩阵因其最优扩散特性在编码理论与对称密钥密码学中扮演核心角色。本文利用斜多项式环 \( \mathbb{F}_q[X;θ,δ] \) 提出一种MDS矩阵构造方法，其中 \( θ\) 是 \( \mathbb{F}_q \) 上的自同构，\( δ\) 是 \( θ\)-导子。我们引入了 \( δ_θ \)-循环矩阵的概念并研究了其结构性质。推导了这些矩阵成为对合矩阵且满足MDS性质的充分必要条件。所得 $δ_θ$-循环矩阵可视为无 $θ$-导子情形下经典构造的推广。本工作的主要贡献之一是准递归MDS矩阵的构造。在斜多项式环 $\mathbb{F}_q[X;θ]$ 框架下，我们构造了与相伴矩阵关联的准递归MDS矩阵。这些矩阵被证明具有对合性，相比文献中先前报道的准对合构造实现了严格改进。文中亦提供了若干说明性结果与示例。