Abstract
In this paper, we introduce the algebraic structure of Z<sub>2</sub> (Z<sub>2</sub>+uZ<sub>2</sub>) (Z<sub>2</sub>+uZ<sub>2</sub>+u<sup>2</sup>Z<sub>2</sub>) -additive codes and Z<sub>2</sub> (Z<sub>2</sub>+uZ<sub>2</sub>) (Z<sub>2</sub>+uZ<sub>2</sub>+u<sup>2</sup>Z<sub>2</sub>) -additive cyclic codes. Compared to the Z<sub>2</sub>Z<sub>4</sub>Z<sub>8</sub>-additive codes, the Gray image of any Z<sub>2</sub> (Z<sub>2</sub>+uZ<sub>2</sub>) (Z<sub>2</sub>+uZ<sub>2</sub>+u<sup>2</sup>Z<sub>2</sub>) -linear code will always be a linear binary code. Therefore, we consider the Z<sub>2</sub> (Z<sub>2</sub>+uZ<sub>2</sub>) (Z<sub>2</sub>+uZ<sub>2</sub>+u<sup>2</sup>Z<sub>2</sub>) -additive cyclic codes as a (Z<sub>2</sub>+uZ<sub>2</sub>+u<sup>2</sup>Z<sub>2</sub>) [x] -submodule of Z<sub>2</sub><sup>α</sup>×(Z<sub>2</sub>+uZ<sub>2</sub>)<sup>β</sup>×(Z<sub>2</sub>+uZ<sub>2</sub>+u<sup>2</sup>Z<sub>2</sub>)<sup>θ</sup>. We give the definition of Z<sub>2</sub> (Z<sub>2</sub>+uZ<sub>2</sub>) (Z<sub>2</sub>+uZ<sub>2</sub>+u<sup>2</sup>Z<sub>2</sub>) -additive codes with generator matrices and parity-check matrices. Furthermore, we give the fundamental result on considering their additive cyclic codes with generator polynomials and spanning sets.
Highlights
Codes over rings were introduced in early 1970s
In this paper, we introduce the algebraic structure of Z2 (Z2+uZ2) (Z2+uZ2+u2Z2) -additive codes and Z2 (Z2+uZ2)
The paper gives the standard form of generator matrix of Z2 RS -additive codes first
Summary
Codes over rings were introduced in early 1970s. this research topic had been widely and really concerned after a milestone paper written by Hammons et al in 1994, which shown that some special classes of non-linear binary codes could be obtained as Gray images of linear codes over. Z2 (Z2 + uZ2 )(Z2 + uZ2 + u2Z2 ) -additive cyclic codes and their generator polynomials and spanning sets are given. Definition 2 A Z2 RS -additive code C of length (α , β ,θ ) is called Z2 RS -additive code of type (α, β ,θ ; k0; k1, k2; k3, k4 , k5), if C is a group isomorphic to the abelian structure. The generator matrix of linear codes can be formed by the minimal spanning set of this linear code. Since the minimum spanning set of the linear codes is not unique, the generator matrix of the linear codes is not unique. In this following, the paper gives the standard form of generator matrix of Z2 RS -additive codes first. The code C ⊥ is of type (α , β ,θ;α − k0; β − k1 − k2 , k2;θ − k3 − k4 − k5 , k5, k4) (11)
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