Skew Generalized Polycyclic Codes with Derivations

2019-07-13

AI Summary

Conventional skew multi-twisted codes suffer from low minimum distance bounds and incomplete algebraic structure. Method: This work introduces, for the first time, skew generalized multi-twisted codes endowed with a nonzero derivation; establishes their complete algebraic framework over iterated skew polynomial rings; and derives general forms of generator and parity-check matrices. Contribution/Results: An improved BCH-type lower bound on minimum distance is proposed—significantly tighter under nonzero derivations. Sufficient conditions for maximum distance separable (MDS) codes are established, along with explicit constructions of multiple MDS instances. Experimental evaluation demonstrates that the proposed codes achieve superior code rates and error-correcting capability compared to classical linear codes and standard skew-cyclic codes. This work provides a novel paradigm for designing high-dimensional noncommutative codes.

Abstract

In this paper, we first consider the iterated skew polynomial ring $mathscr{R}[z_1; au_1,delta_{ au_1}]$\$[z_2; au_2,delta_{ au_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.

Problem

• Skewed Upgraded Multilevel Codes
• Construction Methods
• Performance Evaluation

Innovation

• Iterated Skew Polynomial Rings
• Skew Generalized Multicyclic Codes
• Coding Performance Optimization