Skew Generalized Polycyclic Codes with Derivations

In this paper, we first consider the iterated skew polynomial ring $\mathscr{R}[z_1;\tau_1,\delta_{\tau_1}]$\$[z_2;\tau_2,\delta_{\tau_2}]$, where$\mathscr{R}$is a finite ring with unity. Then we use this structure for the construction of skew generalized polycyclic codes over the ring$\mathscr{R}$and finite field$\mathbb{F}_q$, where$q=p^m$for some positive integer$m$. Further, we derive the structure of the generator and parity check matrices for skew generalized polycyclic codes. Furthermore, we improve the Bose-Chaudhuri-Hocquenghem (BCH) lower bound for a minimum distance of skew generalized polycyclic codes with non-zero derivations over a finite field. Moreover, we find a sufficient condition for a code to be a maximum-distance-separable (MDS) code. In addition, we provide examples of MDS codes to show the importance of our results. A comparative summary of our work with other linear codes is also discussed.