Reversible cyclic codes over finite chain rings

In this paper, necessary and sufficient conditions for the reversibility of a cyclic code of arbitrary length over a finite commutative chain ring have been derived. MDS reversible cyclic codes having length p^s over a finite chain ring with nilpotency index 2 have been characterized and a few examples of MDS reversible cyclic codes have been presented. Further, it is shown that the torsion codes of a reversible cyclic code over a finite chain ring are reversible. Also, an example of a non-reversible cyclic code for which all its torsion codes are reversible has been presented to show that the converse of this statement is not true. The cardinality and Hamming distance of a cyclic code over a finite commutative chain ring have also been determined

On Quantum Codes Obtained From Cyclic Codes Over F2+uF2+u2F2\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2

Let $R=\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$ be a non-chain finite commutative ring, where $u^3=u$. In this paper, we mainly study the construction of quantum codes from cyclic codes over $R$. We obtained self-orthogonal codes over $\mathbb{F}_2$ as gray images of linear and cyclic codes over $R$. The parameters of quantum codes which are obtained from cyclic code over $R$ are discussed.Comment: 9 pages. arXiv admin note: text overlap with arXiv:1407.1232 by other author

Skew constacyclic codes over a non-chain ring Fq[u,v]/⟨f(u),g(v),uv−vu⟩\mathbb{F}_{q}[u,v]/\langle f(u),g(v), uv-vu\rangle

Let $f(u)$ and $g(v)$ be two polynomials of degree $k$ and $\ell$ respectively, not both linear, which split into distinct linear factors over $\mathbb{F}_{q}$. Let $\mathcal{R}=\mathbb{F}_{q}[u,v]/\langle f(u),g(v),\\uv-vu\rangle$ be a finite commutative non-chain ring. In this paper, we study $\psi$-skew cyclic and $\theta_t$-skew constacyclic codes over the ring $\mathcal{R}$ where $\psi$ and $\theta_t$ are two automorphisms defined on $\mathcal{R}$.Comment: 15 page

On Quadratic Residue Codes Over Finite Commutative Chain Rings

Codes over finite rings were initiated in the early 1970s, And they have received much attention after it was proved that important families of binary non-linear codes are images under a Gray map of linear codes over Z4. In this thesis we consider a special families of cyclic codes namely Quadratic residue codes over finite chain rings F2 + uF2 with u2 = 0 and F2 + uF2 + u2F2 with u3 = 0. We study these codes in term of their idempotent generators, and show that these codes have many good properties which are analogous in many respect to properties of Quadratic residue codes over finite fields, also, we study Quadratic residue codes over the ring Z2m, and then generalize this study to Quadratic residue codes over finite commutative chainring Rm-1 = F2 + uF2 + : : : + um-1F2 with um =

Constacyclic Codes Over Finite Principal Ideal Rings

In this paper, we give an important isomorphism between contacyclic codes and cyclic codes over finite principal ideal rings. Necessary and sufficient conditions for the existence of non-trivial cyclic self-dual codes over finite principal ideal rings are given

Multivariable codes over finite chain rings: semisimple codes

The structure of multivariate semisimple codes over a finite chain ring $R$ is established using the structure of the residue field $\bar R$. Multivariate codes extend in a natural way the univariate cyclic and negacyclic codes and include some non-trivial codes over $R$. The structure of the dual codes in the semisimple abelian case is also derived and some conditions on the existence of selfdual codes over $R$ are studied.Comment: Submitted to SIAM Journal on Discrete Mathematic

Repeated-root constacyclic codes over the finite chain ring Fpm[u]/⟨u3⟩\mathbf{ \mathbb{F}_{p^m}[u]/\langle u^3 \rangle }

Let $\mathcal{R}=\mathbb{F}_{p^m}[u]/\langle u^3 \rangle$ be the finite commutative chain ring with unity, where $p$ is a prime, $m$ is a positive integer and $\mathbb{F}_{p^m}$ is the finite field with $p^m$ elements. In this paper, we determine all repeated-root constacyclic codes of arbitrary lengths over $\mathcal{R},$ their sizes and their dual codes. As an application, we list some isodual constacyclic codes over $\mathcal{R}.$ We also determine Hamming distances, RT distances, and RT weight distributions of some repeated-root constacyclic codes over $\mathcal{R}.$Comment: arXiv admin note: text overlap with arXiv:1706.0626

Cyclic codes over some special rings

In this paper we will study cyclic codes over some special rings: F_{q}[u]/(u^{i}), F_{q}[u_1,...u_{i}]/(u_1^2,u_2^2,...,u_{i}^2, u_1 u_2 - u_2 u_1,...,u_{i}u_{j} - u_{j}u_{i},...), F_{q}[u,v]/(u^{i},v^{j},uv-vu), q=p^{r}, where p is a prime number, r\in N-{0} and F_{q} is a field with q elements.Comment: accepted in the Bulletin of the Korean Mathematical Societ

All α+uβ\alpha+u\beta-constacyclic codes of length npsnp^{s} over Fpm+uFpm\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}

Let $\mathbb{F}_{p^{m}}$ be a finite field with cardinality $p^{m}$ and $R=\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}$ with $u^{2}=0$. We aim to determine all $\alpha+u\beta$-constacyclic codes of length $np^{s}$ over $R$, where $\alpha,\beta\in\mathbb{F}_{p^{m}}^{*}$, $n, s\in\mathbb{N}_{+}$ and $\gcd(n,p)=1$. Let $\alpha_{0}\in\mathbb{F}_{p^{m}}^{*}$ and $\alpha_{0}^{p^{s}}=\alpha$. The residue ring $R[x]/\langle x^{np^{s}}-\alpha-u\beta\rangle$ is a chain ring with the maximal ideal $\langle x^{n}-\alpha_{0}\rangle$ in the case that $x^{n}-\alpha_{0}$ is irreducible in $\mathbb{F}_{p^{m}}[x]$. If $x^{n}-\alpha_{0}$ is reducible in $\mathbb{F}_{p^{m}}[x]$, we give the explicit expressions of the ideals of $R[x]/\langle x^{np^{s}}-\alpha-u\beta\rangle$. Besides, the number of codewords and the dual code of every $\alpha+u\beta$-constacyclic code are provided.Comment: arXiv admin note: text overlap with arXiv:1512.01406 by other author

Polyadic cyclic codes over a non-chain ring Fq[u,v]/⟨f(u),g(v),uv−vu⟩\mathbb{F}_{q}[u,v]/\langle f(u),g(v), uv-vu\rangle

Let $f(u)$ and $g(v)$ be any two polynomials of degree $k$ and $\ell$ respectively ($k$ and $\ell$ are not both $1$), which split into distinct linear factors over $\mathbb{F}_{q}$. Let $\mathcal{R}=\mathbb{F}_{q}[u,v]/\langle f(u),g(v),uv-vu\rangle$ be a finite commutative non-chain ring. In this paper, we study polyadic codes and their extensions over the ring $\mathcal{R}$. We give examples of some polyadic codes which are optimal with respect to Griesmer type bound for rings. A Gray map is defined from $\mathcal{R}^n \rightarrow \mathbb{F}^{k\ell n}_q$ which preserves duality. The Gray images of polyadic codes and their extensions over the ring $\mathcal{R}$ lead to construction of self-dual, isodual, self-orthogonal and complementary dual (LCD) codes over $\mathbb{F}_q$. Some examples are also given to illustrate this