# Skew Generalized Polycyclic Codes with Derivations

## TL;DR

This paper introduces a new class of skew generalized polycyclic codes over finite rings and fields, analyzing their structure, minimum distance bounds, and conditions for MDS properties, with practical examples.

## Contribution

It constructs skew generalized polycyclic codes using iterated skew polynomial rings and derives their generator and parity check matrices, also improving the BCH bound.

## Key findings

- Derived the structure of generator and parity check matrices.
- Improved the BCH lower bound for minimum distance.
- Provided conditions and examples for MDS codes.

## Abstract

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.