CYCLIC AND CONSTACYCLIC SELF-DUAL CODES OVER R_k

Karadeniz, Suat,Kelebek, Ismail Gokhan,Yildiz, Bahattin Korean Mathematical Society 2017 대한수학회보 Vol.54 No.4

In this work, we consider constacyclic and cyclic self-dual codes over the rings $R_k$. We start with theoretical existence results for constacyclic and cyclic self-dual codes of any length over $R_k$ and then construct cyclic self-dual codes over $R_1={\mathbb{F}}_2+u{\mathbb{F}}_2$ of even lengths from lifts of binary cyclic self-dual codes. We classify all free cyclic self-dual codes over $R_1$ of even lengths for which non-trivial such codes exist. In particular we demonstrate that our constructions provide a counter example to a claim made by Batoul et al. in [1] and we explain why their claim fails.

On a class of constacyclic codes of length $2p^s$ over $\frac{\mathbb F_{p^m}[u]}{\left\langle u^a \right\rangle}$

Hai Q. Dinh,Bac Trong Nguyen,Songsak Sriboonchitta 대한수학회 2018 대한수학회보 Vol.55 No.4

The aim of this paper is to study the class of $\Lambda$-constacyclic codes of length $2p^s$ over the finite commutative chain ring ${\mathcal R}_a=\frac{\mathbb F_{p^m}[u]}{\left\langle u^a \right\rangle}=\mathbb F_{p^m} + u \mathbb F_{p^m}+ \dots + u^{a-1}\mathbb F_{p^m}$, for all units $\Lambda$ of $\mathcal R_a$ that have the form $\Lambda=\Lambda_0+u\Lambda_1+\dots+u^{a-1}\Lambda_{a-1}$, where $\Lambda_0, \Lambda_1, \dots, \Lambda_{a-1} \in \mathbb F_{p^m}$, $\Lambda_0 \,{\not=}\, 0, \, \Lambda_1 \,{\not=}\, 0$. The algebraic structure of all $\Lambda$-constacyclic codes of length $2p^s$ over ${\mathcal R}_a$ and their duals are established. As an application, this structure is used to determine the Rosenbloom-Tsfasman (RT) distance and weight distributions of all such codes. Among such constacyclic codes, the unique MDS code with respect to the RT distance is obtained.

QUANTUM CODES WITH IMPROVED MINIMUM DISTANCE

Kolotoglu, Emre,Sari, Mustafa Korean Mathematical Society 2019 대한수학회보 Vol.56 No.3

The methods for constructing quantum codes is not always sufficient by itself. Also, the constructed quantum codes as in the classical coding theory have to enjoy a quality of its parameters that play a very important role in recovering data efficiently. In a very recent study quantum construction and examples of quantum codes over a finite field of order q are presented by La Garcia in [14]. Being inspired by La Garcia's the paper, here we extend the results over a finite field with $q^2$ elements by studying necessary and sufficient conditions for constructions quantum codes over this field. We determine a criteria for the existence of $q^2$-cyclotomic cosets containing at least three elements and present a construction method for quantum maximum-distance separable (MDS) codes. Moreover, we derive a way to construct quantum codes and show that this construction method leads to quantum codes with better parameters than the ones in [14].

Quantum codes with improved minimum distance

Emre Kolotouglu,Mustafa Sari 대한수학회 2019 대한수학회보 Vol.56 No.3

The methods for constructing quantum codes is not always sufficient by itself. Also, the constructed quantum codes as in the classical coding theory have to enjoy a quality of its parameters that play a very important role in recovering data efficiently. In a very recent study quantum construction and examples of quantum codes over a finite field of order $q$ are presented by La Garcia in \cite{Guardia}. Being inspired by La Garcia's the paper, here we extend the results over a finite field with $q^2$ elements by studying necessary and sufficient conditions for constructions quantum codes over this field. We determine a criteria for the existence of $q^2$-cyclotomic cosets containing at least three elements and present a construction method for quantum maximum-distance separable (MDS) codes. Moreover, we derive a way to construct quantum codes and show that this construction method leads to quantum codes with better parameters than the ones in \cite{Guardia}.

Cyclic and constacyclic self-dual codes over R_k

Suat Karadeniz,Ismail Gokhan Kelebek,Bahattin Yildiz 대한수학회 2017 대한수학회보 Vol.54 No.4

In this work, we consider constacyclic and cyclic self-dual codes over the rings $R_k$. We start with theoretical existence results for constacyclic and cyclic self-dual codes of any length over $R_k$ and then construct cyclic self-dual codes over $R_1 = \F_2+u\F_2$ of even lengths from lifts of binary cyclic self-dual codes. We classify all free cyclic self-dual codes over $R_1$ of even lengths for which non-trivial such codes exist. In particular we demonstrate that our constructions provide a counter example to a claim made by Batoul et al. in \cite{Batoul} and we explain why their claim fails.

REPEATED-ROOT CONSTACYCLIC CODES OF LENGTH 2p^s OVER GALOIS RINGS

Klin-eam, Chakkrid,Sriwirach, Wateekorn Korean Mathematical Society 2019 대한수학회보 Vol.56 No.1

In this paper, we consider the structure of ${\gamma}$-constacyclic codes of length $2p^s$ over the Galois ring $GR(p^a,m)$ for any unit ${\gamma}$ of the form ${\xi}_0+p{\xi}_1+p^2z$, where $z{\in}GR(p^a,m)$ and ${\xi}_0$, ${\xi}_1$ are nonzero elements of the set ${\mathcal{T}}(p,m)$. Here ${\mathcal{T}}(p,m)$ denotes a complete set of representatives of the cosets ${\frac{GR(p^a,m)}{pGR(p^a,m)}}={\mathbb{F}}p^m$ in $GR(p^a,m)$. When ${\gamma}$ is not a square, the rings ${\mathcal{R}}_p(a,m,{\gamma})=\frac{GR(p^a,m)[x]}{{\langle}x^2p^s-{\gamma}{\rangle}}$ is a chain ring with maximal ideal ${\langle}x^2-{\delta}{\rangle}$, where ${\delta}p^s={\xi}_0$, and the number of codewords of ${\gamma}$-constacyclic code are provided. Furthermore, the self-orthogonal and self-dual ${\gamma}$-constacyclic codes of length $2p^s$ over $GR(p^a,m)$ are also established. Finally, we determine the Rosenbloom-Tsfasman (RT) distances and weight distributions of all such codes.

Repeated-root constacyclic codes of length $2p^s$ over Galois rings

Chakkrid Klin-eam,Wateekorn Sriwirach 대한수학회 2019 대한수학회보 Vol.56 No.1

In this paper, we consider the structure of $\gamma$-constacyclic codes of length $2p^s$ over the Galois ring ${\rm GR}(p^a,m)$ for any unit $\gamma$ of the form $\xi_0+p\xi_1+p^2z$, where $z\in {\rm GR}(p^a,m)$ and $\xi_0, \xi_1$ are nonzero elements of the set $\mathcal{T}(p,m)$. Here $\mathcal{T}(p,m)$ denotes a complete set of representatives of the cosets $\frac{{\rm GR}(p^a,m)}{p{\rm GR}(p^a,m)} = \mathbb{F}_{p^m}$ in ${\rm GR}(p^a,m)$. When $\gamma$ is not a square, the rings $\mathcal{R}_{p}(a,m,\gamma)=\frac{{\rm GR}(p^a,m)[x]}{\langle x^{2p^s}-\gamma\rangle}$ is a chain ring with maximal ideal $\langle x^2-\delta\rangle$, where $\delta^{p^s}=\xi_0$, and the number of codewords of $\gamma$-constacyclic code are provided. Furthermore, the self-orthogonal and self-dual $\gamma$-constacyclic codes of length $2p^s$ over ${\rm GR}(p^a,m)$ are also established. Finally, we determine the Rosenbloom-Tsfasman (RT) distances and weight distributions of all such codes.

SOME CLASSES OF REPEATED-ROOT CONSTACYCLIC CODES OVER Fpm + uFpm + u2Fpm

Xiusheng Liu,Xiaofang Xu 대한수학회 2014 대한수학회지 Vol.51 No.4

Constacyclic codes of length ps over R = Fpm + uFpm + u2Fpm are precisely the ideals of the ring R[x]/(xps −1). In this paper, we investigate constacyclic codes of length ps over R. The units of the ring R are of the forms γ, α+uβ, α+uβ +u2γ and α+u2γ, where α, β and γ are nonzero elements of Fpm . We obtain the structures and Hamming distances of all (α+uβ)-constacyclic codes and (α+uβ+u2γ)-constacyclic codes of length ps over R. Furthermore, we classify all cyclic codes of length ps over R, and by using the ring isomorphism we characterize γ-constacyclic codes of length ps over R.

SOME CLASSES OF REPEATED-ROOT CONSTACYCLIC CODES OVER _p_m+u_p_m+u²_p_m

Liu, Xiusheng,Xu, Xiaofang Korean Mathematical Society 2014 대한수학회지 Vol.51 No.4

Constacyclic codes of length $p^s$ over $R=\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}$ are precisely the ideals of the ring $\frac{R[x]}{<x^{p^s}-1>}$. In this paper, we investigate constacyclic codes of length $p^s$ over R. The units of the ring R are of the forms ${\gamma}$, ${\alpha}+u{\beta}$, ${\alpha}+u{\beta}+u^2{\gamma}$ and ${\alpha}+u^2{\gamma}$, where ${\alpha}$, ${\beta}$ and ${\gamma}$ are nonzero elements of $\mathbb{F}_{p^m}$. We obtain the structures and Hamming distances of all (${\alpha}+u{\beta}$)-constacyclic codes and (${\alpha}+u{\beta}+u^2{\gamma}$)-constacyclic codes of length $p^s$ over R. Furthermore, we classify all cyclic codes of length $p^s$ over R, and by using the ring isomorphism we characterize ${\gamma}$-constacyclic codes of length $p^s$ over R.

SKEW CONSTACYCLIC CODES OVER FINITE COMMUTATIVE SEMI-SIMPLE RINGS

Dinh, Hai Q.,Nguyen, Bac Trong,Sriboonchitta, Songsak Korean Mathematical Society 2019 대한수학회보 Vol.56 No.2

This paper investigates skew ${\Theta}-{\lambda}$-constacyclic codes over $R=F_0{\oplus}F_1{\oplus}{\cdots}{\oplus}F_{k-1}$, where $F{_i}^{\prime}s$ are finite fields. The structures of skew ${\lambda}$-constacyclic codes over finite commutative semi-simple rings and their duals are provided. Moreover, skew ${\lambda}$-constacyclic codes of arbitrary length are studied under a new definition. We also show that a skew cyclic code of arbitrary length over finite commutative semi-simple rings is equivalent to either a cyclic code over R or a quasi-cyclic code over R.