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

Abstract

In this paper, we study a class of skew-cyclic codes using a skew polynomial ring over R = Z 4 + uZ 4 ; u 2 = 1, with an automorphism θ and a derivation δ θ. We generalize the notion of cyclic codes to skew-cyclic codes with derivation, and call such codes as δ θ-cyclic codes. Some properties of skew polynomial ring R[x, θ, δ θ ] are presented. A δ θ-cyclic code is proved to be a left R[x, θ, δ θ ]-submodule of R[x,θ,δ θ ] x n −1. The form of a parity-check matrix of a free δ θ-cyclic codes of even length n is presented. These codes are further generalized to double δ θ-cyclic codes over R. We have obtained some new good codes over Z 4 via Gray images and residue codes of these codes. The new codes obtained have been reported and added to the database of Z 4-codes [2].

Skew cyclic codes over $\mathbb{Z}_4+v\mathbb{Z}_4$ with derivation

“…This observation brings us to the conclusion that the codes Res(C k ), Tor(C k ), and Φ(C k ) has at most 4 k , 4 k , and 4 2k codewords, respectively. Moreover, the similar observation can also be applied to the code over Z 4 + uZ 4 , with u 2 = 1, investigated by Sharma and Bhaintwal [23] and it is easy to verify that in this case, the codes Res(C k ), Tor(C k ), and Φ(C k ) has at most 4 2k codewords.…”

“…They [9] also constructed MDS as well as MRD codes from certain families of the skew cyclic codes. Sharma and Bhaintwal [23] extended the study of these skew cyclic codes over a finite ring, namely over the ring Z 4 + uZ 4 , with u 2 = 1. Via residue codes, Plotkin sum, or the Gray map they [23] defined, several linear codes over Z 4 with good parameters were obtained.…”

“…Sharma and Bhaintwal [23] extended the study of these skew cyclic codes over a finite ring, namely over the ring Z 4 + uZ 4 , with u 2 = 1. Via residue codes, Plotkin sum, or the Gray map they [23] defined, several linear codes over Z 4 with good parameters were obtained. Continuing the study of [23], in this paper, we investigate a class of skew cyclic codes with derivation over the ring Z 4 + vZ 4 , with v 2 = v.…”

“…Via residue codes, Plotkin sum, or the Gray map they [23] defined, several linear codes over Z 4 with good parameters were obtained. Continuing the study of [23], in this paper, we investigate a class of skew cyclic codes with derivation over the ring Z 4 + vZ 4 , with v 2 = v.…”

A class of constacyclic codes and skew constacyclic codes over $$\pmb {\mathbb {Z}}_{2^s}+u\pmb {\mathbb {Z}}_{2^s}$$ and their gray images

“…[25] studied constacyclic codes over Z 4 [u] u 2 −1 , and obtained some optimal codes over Z 4 using Gray maps. Many authors have studied cyclic and constacyclic codes over Z 4 and its extensions and obtained optimal quaternary codes or optimal binary codes via Gray maps [21][22][23]38].…”