Skew Generalized Polycyclic Codes with Derivations

Shikha Patel; Om Prakash

paper

Skew Generalized Polycyclic Codes with Derivations

arXiv:1907.06086

Abstract

In this paper, we first consider the iterated skew polynomial ring $R [z_{1}; I ¨ _{1}, I^´_{I ¨ _{1}}]$ \$[z_2;Ï_2,Î´_{Ï_2}] $, w h er e$ \mathscr{R} $i s a f ini t er in g w i t h u ni t y . T h e n w e u se t hi ss t r u c t u r e f or t h eco n s t r u c t i o n o f s k e w g e n er a l i z e d p o l y cy c l i cco d eso v er t h er in g$ \mathscr{R} $an df ini t e f i e l d$ \mathbb{F}_q $, w h er e$ q=p^m $f or so m e p os i t i v e in t e g er$ 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.

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