Skew Generalized Polycyclic Codes with Derivations
Shikha Patel, Om Prakash
TL;DR
The paper addresses constructing two-dimensional skew generalized polycyclic codes over an iterated skew polynomial ring $B= \\mathscr{R}[z_1;\\tau_1,\\delta_{\\tau_1}][z_2;\\tau_2,\\delta_{\\tau_2}]$, deriving their generator structures and distance bounds. It proves that skew generalized polycyclic codes are left $B$-submodules via a map $\\Gamma_f$ and are free $\\mathbb{F}_q$-modules with dimension $k=(l-k_1)(s-k_2)$, with a basis generated from $g(z_1,z_2)$. The work provides explicit generator matrices, characterizes dual codes, extends BCH-type lower bounds for non-zero derivations, and gives a sufficient condition for MDS codes along with several MDS examples. Compared to classical cyclic, polycyclic, and two-dimensional codes, the results demonstrate improved parameter flexibility and practical code constructions, as validated by Magma-based examples.
Abstract
In this paper, we first consider the iterated skew polynomial ring $\mathscr{R}[z_1;τ_1,δ_{τ_1}]$\\$[z_2;τ_2,δ_{τ_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.