Bounds and Equivalence of Skew Polycyclic Codes over Finite Fields

We study skew polycyclic codes over a finite field $\mathbb{F}_q$, associated with a skew polynomial $f(x) \in \mathbb{F}_q[x;\sigma]$, where $\sigma$ is an automorphism of $\mathbb{F}_q$. We start by proving the Roos-like bound for both the Hamming and the rank metric for this class of codes. Next, we focus on the Hamming and rank equivalence between two classes of polycyclic codes by introducing an equivalence relation and describing its equivalence classes. Finally, we present examples that illustrate applications of the theory developed in this paper.