Construction of skew cyclic codes over $\mathbb F_q+v\mathbb F_q$

Abstract

In this paper skew cyclic codes over the the family of rings Fq +vFq with v 2 = v are studied for the first time in its generality. Structural properties of skew cyclic codes over Fq + vFq are investigated through a decomposition theorem. It is shown that skew cyclic codes over this ring are principally generated. The idempotent generators of skew-cyclic codes over Fq and Fq +vFq have been considered for the first time in literature. Moreover, a BCH type bound is presented for the parameters of these codes.

Skew Cyclic Codes over a Non‐chain Ring

“…The proof of the following theorem is similar to that of (Ref. [8], Theorem 6), so we omit the proof here.…”

“…Lemma 3 (Ref. [8], Lemma 2) Let g(x) ∈ F q [x, θ k ] be a monic right divisor of x n − 1. If (n, t k ) = 1, then…”

“…[6], T. Abualrub and P. Seneviratne studied skew cyclic codes over the ring F 2 + vF 2 with v 2 = v by defining the automorphism θ as: [7], Gao investigated the skew cyclic codes over F p + vF p with v 2 = v. In the recent work of Ref. [8], Gursoy et al studied the structural properties of skew cyclic codes over F q + vF q with v 2 = v, which used a different automorphism from the one given in Ref. [7].…”

$\mathbb{Z}_{q}(\mathbb{Z}_{q}+u\mathbb{Z}_{q})-$ linear skew constacyclic codes

“…have been studied in [1,2,5,13,21,23,26,30,32,33] as a generalization of skew cyclic codes over finite fields. Recently, P. Li et al [28] gave the structure of (1 + u)-constacyclic codes over the ring Z 2 Z 2 [u] and Aydogdu et al [6] studied Z 2 Z 2 [u]-cyclic and constacyclic codes.…”

A class of skew-cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ with derivation

“…They have been further generalized in many ways [8,10,11]. In recent years many researchers have shown interest in this direction [4,21,16,14], and many new results on codes over different rings in the setting of skew polynomial rings have been obtained. However, almost all this work has been done in the setting of skew-polynomial rings with automorphism only.…”