Related papers: $\mathbb{Z}_p\mathbb{Z}_p[u]$-additive codes

In this paper, two different Gray-like maps from $Z_p^\alpha\times Z_{p^k}^\beta$, where $p$ is prime, to $Z_p^n$, $n={\alpha+\beta p^{k-1}}$, denoted by $\phi$ and $\Phi$, respectively, are presented. We have determined the connection…

In this paper, we mainly study the theory of linear codes over the ring $R =\mathbb{Z}_4+u\mathbb{Z}_4+v\mathbb{Z}_4+uv\mathbb{Z}_4$. By the Chinese Remainder Theorem, we have $R$ is isomorphic to the direct sum of four rings…

Let $\mathbb{Z}_{p}$ be the ring of residue classes modulo a prime $p$. The $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic codes of length $(\alpha,\beta)$ is identify as $\mathbb{Z}_{p}[u,v][x]$-submodule of $\mathbb{Z}_{p}[x]/\langle…

In this paper, we study the algebraic structure of Z_2[u]Z_2[u, v]-additive codes which are Z_2[u, v]-submodules where u^2 = v^2 = 0 and uv = vu. In particular, we determine a Gray map from Z_2[u]Z_2 [u, v] to Z_2^{2{\alpha}+8\b{eta}} and…

A code $C = \Phi(\mathcal{C})$ is called $\mathbb{Z}_p \mathbb{Z}_{p^2}$-linear if it's the Gray image of the $\mathbb{Z}_p \mathbb{Z}_{p^2}$-additive code $\mathcal{C}$. In this paper, the rank and the dimension of the kernel of…

In this paper, we introduce a new definitions of the Gray weight and the Gray map for linear codes over $\mathbb{Z}_9+u\mathbb{Z}_9$ with $u^2=u$. Some results on self-dual codes over this ring are investigated. Further, the structural…

In this paper, we study a relative two-weight $\mathbb{Z}_2 \mathbb{Z}_4$-additive codes. It is shown that the Gray image of a two-distance $\mathbb{Z}_2 \mathbb{Z}_4$-additive code is a binary two-distance code and that the Gray image of a…

Let $\mathcal{R}=\mathbb{F}_{p}+u\mathbb{F}_{p}+u^2\mathbb{F}_{p}+u^3\mathbb{F}_{p}$ with $u^4=u$ be a finite non-chain ring, where $p$ is a prime congruent to $1$ modulo $3$. In this paper we study $(1-2u^3)$-constacyclic codes over the…

In the paper, we firstly study the algebraic structures of $\mathbb{Z}_p \mathbb{Z}_{p^k}$-additive cyclic codes and give the generator polynomials and the minimal spanning set of these codes. Secondly, a necessary and sufficient condition…

In this paper, we study one-Lee weight and two-Lee weight codes over $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$, where $u^{2}=0$. Some properties of one-Lee weight $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes are given, and a complete…

Let C be an additive subgroup of $\Z_{2k}^n$ for any $k\geq 1$. We define a Gray map $\Phi:\Z_{2k}^n \longrightarrow \Z_2^{kn}$ such that $\Phi(\codi)$ is a binary propelinear code and, hence, a Hamming-compatible group code. Moreover,…

Let $R=\mathbb{Z}_4+u\mathbb{Z}_4,$ where $\mathbb{Z}_4$ denotes the ring of integers modulo $4$ and $u^2=0$. In the present paper, we introduce a new Gray map from $R^n$ to $\mathbb{Z}_{4}^{2n}.$ We study $(1+2u)$-constacyclic codes over…

The $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-additive codes are subgroups of $\mathbb{Z}_p^{\alpha_1} \times \mathbb{Z}_{p^2}^{\alpha_2} \times \cdots \times \mathbb{Z}_{p^s}^{\alpha_s}$, and can be seen as linear codes over…

Let $m\geq 2$ be any natural number and let $\mathcal{R}=\mathbb{F}_{p}+u\mathbb{F}_{p}+u^2\mathbb{F}_{p}+\cdots+u^{m-1}\mathbb{F}_{p}$ be a finite non-chain ring, where $u^m=u$ and $p$ is a prime congruent to $1$ modulo $(m-1)$. In this…

Let $p$ be a prime number and $\varsigma$ and $m$ be a positive integers. Let $\mathcal{R} = \mathbb{F}_{2^m} + u\mathbb{F}_{2^m} + u^2\mathbb{F}_{2^m}$ ($u^3 = 0$). Cyclic codes of length $2^\varsigma$ over $\mathcal{R}$ are precisely the…

Let $\mathcal{A}$ and $\mathcal{B}$ be two prime $C^{*}$-algebras. In this paper, we investigate the additivity of map $\Phi$ from $\mathcal{A}$ onto $\mathcal{B}$ that are bijective unital and satisfies $$\Phi(AP+\lambda…

We first define a new Gray map from $R=\mathbb{Z}_4+u\mathbb{Z}_4$ to $\mathbb{Z}^{2}_{4}$, where $u^2=1$ and study $(1+2u)$-constacyclic codes over $R$. Also of interest are some properties of $(1+2u)$-constacyclic codes over $R$.…

We investigate additive cyclic codes over the alphabet $\mathbb{F}_{q}\mathbb{F}_{q^2}$, where $q$ is a prime power. First, its generator polynomials and minimal spanning set are determined. Then, examples of $\mathbb{F}_{q^2}$-additive…

For any positive integer $m$ and an odd prime $p$; let $\mathbb{F}_{q}+u\mathbb{F}_{q}$, where $q=p^{m}$, be a ring extension of the ring $\mathbb{F}_{p}+u\mathbb{F}_{p}.$ In this paper, we construct linear codes over…

The $\Z_p\Z_{p^2}$-additive codes are subgroups of $\Z_p^{\alpha_1} \times \Z_{p^2}^{\alpha_2}$, and can be seen as linear codes over $\Z_p$ when $\alpha_2=0$, $\Z_{p^2}$-additive codes when $\alpha_1=0$, or $\Z_2\Z_4$-additive codes when…