MDS matrices from skew polynomials with automorphisms and derivations

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 $\mathbb{F}_q[X;\theta,\delta]$, where $\theta$ is an automorphism and $\delta$ is a $\theta$-derivation on $\mathbb{F}_q$. We introduce the notion of $\delta_{\theta}$-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 $\delta_\theta$-circulant matrix can be viewed as a generalization of classical constructions obtained in the absence of $\theta$-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 $\mathbb{F}_q[X;\theta]$, 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.