Related papers: Complete classification for simple root cyclic cod…

Let $R=\mathbb{Z}_{4}[v]/\langle v^2+2v\rangle=\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ ($v^2=2v$) and $n$ be an odd positive integer. Then $R$ is a local non-principal ideal ring of $16$ elements and there is a $\mathbb{Z}_{4}$-linear Gray map from…

Let $p$ be a prime number. In this paper, we study cyclic codes over the ring $ \Z_p[u, v]/\langle u^2, v^2, uv-vu\rangle$. We find a unique set of generators for these codes. We also study the rank and the Hamming distance of these codes.…

Let $R=\mathbb{Z}_{4}[u]/\langle u^k\rangle=\mathbb{Z}_{4}+u\mathbb{Z}_{4}+\ldots+u^{k-1}\mathbb{Z}_{4}$ ($u^k=0$) where $k\in \mathbb{Z}^{+}$ satisfies $k\geq 2$. For any odd positive integer $n$, it is known that cyclic codes over $R$ of…

Let $R$ be a finite commutative chain ring with unique maximal ideal $\langle \gamma\rangle$, and let $n$ be a positive integer coprime with the characteristic of $R/\langle \gamma\rangle$. In this paper, the algebraic structure of cyclic…

Let $\mathbb{F}_{2^m}$ be a finite field of characteristic $2$ and $R=\mathbb{F}_{2^m}[u]/\langle u^k\rangle=\mathbb{F}_{2^m} +u\mathbb{F}_{2^m}+\ldots+u^{k-1}\mathbb{F}_{2^m}$ ($u^k=0$) where $k\in \mathbb{Z}^{+}$ satisfies $k\geq 2$. For…

Let $r,s,t$ be three positive integers and $\mathcal{C}$ be a binary linear code of lenght $r+s+t$. We say that $\mathcal{C}$ is a triple cyclic code of lenght $(r,s,t)$ over $\mathbb{Z}_2$ if the set of coordinates can be partitioned into…

Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$, $R=\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ $(u^2=0)$ and $s,n$ be positive integers such that $n$ is odd. In this paper, we give an explicit representation for every self-dual…

In this paper, we introduce $\mathbb{Z}_{p^r}\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive cyclic codes for $r\leq s$. These codes can be identified as $\mathbb{Z}_{p^s}[x]$-submodules of $\mathbb{Z}_{p^r}[x]/\langle x^{\alpha}-1\rangle \times…

Let $p$ be a prime number. In this paper, we discuss the structures of cyclic codes over the ring $ \mathbb{F}_p[u, v] / \langle u^k, v^2, uv-vu\rangle$. We find a unique set of generators for these codes. We also study the rank and the…

In this paper, we investigate cyclic codes over the ring $ \mathbb{F}_p[u,v,w]\langle u^2,$ $v^2, w^2$, $uv-vu, vw-wv, uw-wu \rangle$, where $p$ is a prime number. Which is a part of family of Frobenius rings. We find a unique set of…

In this paper, we describe linear and cyclic codes over the rings of the form $R_{s,p}=\mathbb{Z}_{p}[u]/\left( f\left(u\right) /\left( u-s\right) \right)$, where $p$ is a prime number and $f\left( u\right) =u^{p}-u$, with $s\in…

Let $R=\mathbb{Z}_4$ be the integer ring mod $4$. A double cyclic code of length $(r,s)$ over $R$ is a set that can be partitioned into two parts that any cyclic shift of the coordinates of both parts leaves invariant the code. These codes…

For odd length $n$, the cyclic codes construction over $\Re= \Z_4[v]/ \langle v^2-v \rangle$ is provided. The hulls of cyclic codes over $\Re$ are studied. The average $2$-dimension $E(n)$ of the hulls of cyclic codes over $\Re$ is also…

Cyclic codes over R have been introduced recently. In this paper, we study the cyclic codes over R and their $\Z_2$ image. Making use of algebraic structure, we find the some good $\Z_2$ codes of length 28.

In this paper, we study skew cyclic codes with arbitrary length over the ring $R=\mathbb{F}_{p}+u\mathbb{F}_{p}$ where $p$ is an odd prime and $% u^{2}=0$. We characterize all skew cyclic codes of length $n$ as left $% R[x;\theta…

Self-dual and complementary dual cyclic/abelian codes over finite fields form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, abelian codes…

Self-dual cyclic codes form an important class of linear codes. It has been shown that there exists a self-dual cyclic code of length $n$ over a finite field if and only if $n$ and the field characteristic are even. The enumeration of such…

In this paper, we investigate cyclic code over the ring $\mathbb{F}_{p^k} + v\mathbb{F}_{p^k} + v^2\mathbb{F}_{p^k} + ... + v^r\mathbb{F}_{p^k}$, where $v^{r+1}=v$, $p$ a prime number, $r>1$ and $\gcd(r,p)=1$, we prove as generalisation of…

Cyclic codes, as a crucial subclass of linear codes, exhibit broad applications in communication systems, data storage systems, and consumer electronics, primarily attributed to their well-structured algebraic properties. Let $p$ denote an…

This paper presents some basic theorems giving the structure of cyclic codes of length n over the ring of integers modulo p^a and over the p-adic numbers, where p is a prime not dividing n. An especially interesting example is the 2-adic…