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In this paper we present a class of 2D skew-cyclic codes over R= Fq+ uFq, u2= 1 , using the bivariate skew polynomial ring R[x, y, θ, σ] , where Fq is a finite field, and θ and σ are two commuting automorphisms of R. After defining a partial order on R[x, y, θ, σ] , we obtain division algorithm for R[x, y, θ, σ] under two different conditions. The structure of 2D skew-cyclic codes over R is obtained in terms of their generating sets. For this, we have classified these codes into different classes, based on certain conditions they satisfy, and accordingly obtained their generating sets in each case separately. A decomposition of a 2D skew-cyclic code C over R into 2D skew-cyclic codes over Fq is studied and some examples are given to illustrate the results. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

Department of Mathematics

School of Advanced Sciences

Vellore Campus

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Applicable Algebra in Engineering, Communications and Computing

A class of 2D skew-cyclic codes over Fq+ uFq

VIT Unitversity

Electrical

School of Electrical Engineering

Chennai Campus

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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 〈xn −1 〉 θ] . 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]. © 2018 AIMS.

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A class of skew-cyclic codes over ℤ 4 + uℤ 4 with derivation

In this paper, we study cyclic codes over R=4+u4,u2 = 0. A necessary and sufficient condition for a cyclic code over R to contain its dual is determined. The odd and even length cases are discussed separately to obtain above condition. It is shown that Gray image of a cyclic code over R containing its dual is a linear code over 4 which also contains its dual. We have then obtained the parameters of corresponding CSS-quantum codes over 4. By augmentation, we construct codes with dual-containing property from codes of smaller size containing their duals. Through this construction, we have obtained some optimal quantum codes over 4. Some examples have been given to illustrate the results. © 2018 World Scientific Publishing Company.

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In this paper, we study skew-cyclic codes over the ring R = GR(4, 2) + uGR(4, 2), u2 = u, where GR(4, 2) is the Galois extension of ℤ4 of degree 2. We describe some structural properties of skew polynomial ring R[x, θ], where θ is an automorphism of R. A sufficient condition for skew cyclic codes over R to be free is presented. It is shown that skew-cyclic codes over R are either equivalent to cyclic codes or to quasi-cyclic codes. A brief description of the duals of these codes is also presented. We define a Gray map from R to [GR(4, 2)]2, and show that the Gray image of a skew-cyclic codes over R is a skew 2-quasi cyclic code over GR(4, 2). © 2015 IEEE.

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