Let q be a prime power and F_q be a finite field. In this paper, we study constacyclic codes over the ring F_q+ u F_q +v F_q+ u v F_q, where u^2=u, v^2=v and uv=vu. We characterized the generator polynomials of constacyclic codes and their duals using some decomposition of this ring. We also define a gray map and characterize the Gray images of self-dual cyclic codes over F_q+uF_q+vF_q+uvF_q.
Constacyclic codes over finite fields are a family of linear codes and contain cyclic codes as a subclass. Constacyclic codes are related to many areas of mathematics and outperform cyclic codes in several aspects. Hence, constacyclic codes are of theoretical importance. On the other hand, constacyclic codes are important in practice, as they have rich algebraic structures and may have efficient decoding algorithms. In this paper, two classes of constacyclic codes are constructed using a general construction of constacyclic codes with cyclic codes. The first class of constacyclic codes is moti…
In this paper we introduce the notion of $λ$-constacyclic codes over finite rings $R$ for arbitary element $λ$ of $R$. We study the non-invertible-element constacyclic codes (NIE-constacyclic codes) over finite principal ideal rings (PIRs). We determine the algebraic structures of all NIE-constacyclic codes over finite chain rings, give the unique form of the sets of the defining polynomials and obtain their minimum Hamming distances. A general form of the duals of NIE-constacyclic codes over finite chain rings is also provided. In particular, we give a necessary and sufficient condition for t…
Recently, entanglement-assisted quantum error correcting codes (EAQECCs) have been constructed by cyclic codes and negacyclic codes. In this paper, by analyzing the cyclotomic cosets in the defining set of constacyclic codes, we constructed three classes of new EAQECCs which satisfy the entanglement-assisted quantum Singleton bound. Besides, three classes of EAQECCs with maximal entanglement from constacyclic codes are constructed in the meanwhile.
In this paper skew constacyclic codes over finite non-chain ring R = F_q+uF_q+vF_q, where q= p^m, p is an odd prime and u^{2}=u, v^{2}=v, uv = vu = 0 are studied. We show that Gray image of a skew alpha-constacyclic cyclic code of length n over R is a skew alpha-quasi-cyclic code of length $3n$ over F_{q} of index 3. It is also shown that skew alpha-constacyclic codes are either equivalent to alpha-constacyclic codes or alpha-quasi-twisted codes over R. Further, the structural properties of skew constacyclic over R are obtained by decomposition method.
Recently, a new algorithm to test equivalence of two cyclic codes has been introduced which is efficient and produced useful results. In this work, we generalize this algorithm to constacyclic codes. As an application of the algorithm we found many constacyclic codes with good parameters and properties. In particular, we found 23 new codes that improve the minimum distances of BKLCs.
Multi-dimensional cyclic code is a natural generalization of cyclic code. In an earlier paper we explored two-dimensional constacyclic codes over finite fields. Following the same technique, here we characterize the algebraic structure of multi-dimensional constacyclic codes, in particular three-dimensional $(α,β,γ)$- constacyclic codes of arbitrary length $s\ell k$ and their duals over a finite field $\mathbb{F}_q$, where $α,β,γ$ are non zero elements of $\mathbb{F}_q$. We give necessary and sufficient conditions for a three-dimensional $(α,β,γ)$- constacyclic code to be self-dual.
In this paper, several families of irreducible constacyclic codes over finite fields and their duals are studied. The weight distributions of these irreducible constacyclic codes and the parameters of their duals are settled. Several families of irreducible constacyclic codes with a few weights and several families of optimal constacyclic codes are constructed. As by-products, a family of $[2n, (n-1)/2, d \geq 2(\sqrt{n}+1)]$ irreducible cyclic codes over $\gf(q)$ and a family of $[(q-1)n, (n-1)/2, d \geq (q-1)(\sqrt{n}+1)]$ irreducible cyclic codes over $\gf(q)$ are presented, where $n$ is a …
Construction of good quantum codes via classical codes is an important task for quantum information and quantum computing. In this work, by virtue of a decomposition of the defining set of constacyclic codes we have constructed eight new classes of entanglement-assisted quantum maximum distance separable codes.
In this paper we characterize the algebraic structure of two-dimensional $(α,β)$-constacyclic codes of arbitrary length $s.\ell$ and of their duals. For $α,β\in \{1,-1\}$, we give necessary and sufficient conditions for a two-dimensional $(α,β)$-constacyclic code to be self-dual. We also show that a two-dimensional $(α,1 )$-constacyclic code $\mathcal{C}$ of length $n=s.\ell$ can not be self-dual if $\gcd(s,q)= 1$. Finally, we give some examples of self-dual, isodual, MDS and quasi-twisted codes corresponding to two-dimensional $(α,β)$-constacyclic codes.
In this note, we study skew cyclic and skew constacyclic codes over the ring $\mathcal{R}=F_{q}+uF_{q}+vF_{q}+uvF_{q}$ where $q=p^{m},$ $p$ is an odd prime, $u^{2}=u,~v^{2}=v,~uv=vu$. We show that Gray images of a skew cyclic and skew $α$-constacyclic code of length $n$ are skew quasi-cyclic code of length $4n$ over $F_{q}$ of index 4. Also, it is shown that skew $α$-constacyclic codes are either equivalent to $α$-constacyclic codes or $α$-quasi-twisted codes over $\mathcal{R}$. Further, structural properties, specially, generating polynomials and idempotent generators for skew cyclic and skew…
Maximum distance separable (MDS) codes are optimal in the sense that the minimum distance cannot be improved for a given length and code size. The most prominent MDS codes are generalized Reed-Solomon (GRS) codes. The square $\mathcal{C}^{2}$ of a linear code $\mathcal{C}$ is the linear code spanned by the component-wise products of every pair of codewords in $\mathcal{C}$. For an MDS code $\mathcal{C}$, it is convenient to determine whether $\mathcal{C}$ is a GRS code by determining the dimension of $\mathcal{C}^{2}$. In this paper, we investigate under what conditions that MDS constacyclic c…
In this paper, we study the $σ$-self-orthogonality of constacyclic codes of length $p^s$ over the finite commutative chain ring $\mathbb F_{p^m} + u \mathbb F_{p^m}$, where $u^2=0$ and $σ$ is a ring automorphism of $\mathbb F_{p^m} + u \mathbb F_{p^m}$. First, we obtain the structure of $σ$-dual code of a $λ$-constacyclic code of length $p^s$ over $\mathbb F_{p^m} + u \mathbb F_{p^m}$. Then, the necessary and sufficient conditions for a $λ$-constacyclic code to be $σ$-self-orthogonal are provided. In particular, we determine the $σ$-self-dual constacyclic codes of length $p^s$ over $\mathbb F_…
A code is called a locally repairable code (LRC) if any code symbol is a function of a small fraction of other code symbols. When a locally repairable code is employed in a distributed storage systems, an erased symbol can be recovered by accessing only a small number of other symbols, and hence alleviating the network resources required during the repair process. In this paper we consider repeated-root constacyclic codes, which is a generalization of cyclic codes, that are optimal with respect to a Singleton-like bound on minimum distance. An LRC with the structure of a constacyclic code can …
In this paper, we study skew constacyclic codes over the ring $\mathbb{Z}_{q}R$ where $R=\mathbb{Z}_{q}+u\mathbb{Z}_{q}$, $q=p^{s}$ for a prime $p$ and $u^{2}=0$. We give the definition of these codes as subsets of the ring $\mathbb{Z}_{q}^αR^β$. Some structural properties of the skew polynomial ring $ R[x,θ]$ are discussed, where $ θ$ is an automorphism of $R$. We describe the generator polynomials of skew constacyclic codes over $ R $ and $\mathbb{Z}_{q}R$. Using Gray images of skew constacyclic codes over $\mathbb{Z}_{q}R$ we obtained some new linear codes over $\mathbb{Z}_4$. Further, we h…
Many kinds of codes which possess two cycle structures over two special finite commutative chain rings, such as ${\Bbb Z}_2{\Bbb Z}_4$-additive cyclic codes and quasi-cyclic codes of fractional index etc., were proved asymptotically good. In this paper we extend the study in two directions: we consider any two finite commutative chain rings with a surjective homomorphism from one to the other, and consider double constacyclic structures. We construct an extensive kind of double constacyclic codes over two finite commutative chain rings. And, developing a probabilistic method suitable for quasi…
Quantum maximum-distance-separable (MDS) codes are an important class of quantum codes. In this paper, using constacyclic codes and Hermitain construction, we construct some new quantum MDS codes of the form $q=2am+t$, $n=\frac{q^{2}+1}{a}$. Most of these quantum MDS codes are new in the sense that their parameters are not covered be the codes available in the literature.
The aim of this paper is to give conditions for the equivalency between skew constacyclic codes, skew cyclic codes and skew negacyclic codes defined over semi-local rings. Also, we provide construction and an enumeration of Euclidean and Hermitian skew LCD cyclic codes over $ \mathbb{F}_{p^{t}}+ v \mathbb{F}_{p^{t}} $. Several optimal LCD codes are obtained from Gray images of LCD skew codes over semi-local rings.
In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring $\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangle$ where $p$ is an odd prime and $m \geq 1$. In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring $\mathcal{R}$. Then a necessary and sufficient condition for skew constacyclic codes over $\mathcal{R}$ to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new n…
On constacyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$ and their Gray images
首先定义了一个新的从$R=\mathbb{Z}_4+u\mathbb{Z}_4$到$\mathbb{Z}^{2}_{4}$的Gray映射，其中$u^2=1$和研究了$R$上的$(1+2u)$ -常循环码.我们还研究了$R$上的$(1+2u)$ -常循环码的一些性质.考虑它们的$\mathbb{Z}_4$象，我们证明了$(1+2u)$ -长度为$n$ over $R$的常循环码的Gray象是长度为$2n$ over $\mathbb{Z}_4$的循环码.在许多情况下，后者的代码有更好的参数比Aydin和Asamov的在线数据库。我们还给出了Özen等人在Finite Fields and Their Applications，{ \bf 38}，（2016）27-39中发表的新循环$R$ -码表的修正版本。
Entanglement-assisted quantum error correcting codes (EAQECCs) can be derived from arbitrary classical linear codes. However, it is a very difficult task to determine the number of entangled states required. In this work, using the method of the decomposition of the defining set of constacyclic codes, we construct two families of q-ary entanglement-assisted quantum MDS (EAQMDS) codes based on classical constacyclic MDS codes by exploiting less pre-shared maximally entangled states. We show that a class of q-ary EAQMDS have minimum distance upper bound greater than q. Some of them have much lar…
One of the most important and challenging problems in coding theory is to determine the optimal values of the parameters of a linear code and to explicitly construct codes with optimal parameters, or as close to the optimal values as possible. The class of quasi-twisted (QT) codes has been very promising in this regard. Over the past few decades various search algorithms to construct QT codes with better parameters have been employed. Most of these algorithms (such as ASR) start by joining constacyclic codes of smaller lengths to obtain QT codes of longer lengths. There has been an algorithm t…
For any constacyclic code over a finite commutative chain ring of length coprime to the characteristic of the ring, we construct explicitly generator polynomials and check polynomials, and exhibit a BCH bound for such constacyclic codes. As a consequence, such constacyclic codes are principal. Further, we get a necessary and sufficient condition that the cyclic codes over a finite commutative principal ideal ring are all principal. This condition is still sufficient for constacyclic codes over such rings being principal.
We introduce a Gray map from $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ to $\mathbb{F}_{2}^{2m}$ and study $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}},$ where $u^{2}=0.$ It is proved that the image of a $(1+u)$-constacyclic code length $n$ over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ under the Gray map is a distance-invariant quasi-cyclic code of index $m$ and length $2mn$ over $\mathbb{F}_{2}.$ We also prove that every code of length $2mn$ which is the Gray image of cyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ of length $n$ is permutation equivalent to…
Let $m,e$ be positive integers, $p$ a prime number, $\mathbb{F}_{p^m}$ be a finite field of $p^m$ elements and $R=\mathbb{F}_{p^m}[u]/\langle u^e\rangle$ which is a finite chain ring. For any $ω\in R^\times$ and positive integers $k, n$ satisfying ${\rm gcd}(p,n)=1$, we prove that any $(1+ωu)$-constacyclic code of length $p^kn$ over $R$ is monomially equivalent to a matrix-product code of a nested sequence of $p^k$ cyclic codes with length $n$ over $R$ and a $p^k\times p^k$ matrix $A_{p^k}$ over $\mathbb{F}_p$. Using the matrix-product structures, we give an iterative construction of every $(1…
The entanglement-assisted (EA) formalism allows arbitrary classical linear codes to transform into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this work, we propose a decomposition of the defining set of constacyclic codes. Using this method, we construct four classes of $q$-ary entanglement-assisted quantum MDS (EAQMDS) codes based on classical constacyclic MDS codes by exploiting less pre-shared maximally entangled states. We show that a class of $q$-ary EAQMDS have minimum distance upper limit greate…
Let $\mathbb{F}_p$ be a finite field and $u$ be an indeterminate. This article studies $(1-2u^k)$-constacyclic codes over the ring $\mathcal{R}=\mathbb{F}_p+u\mathbb{F}_p+u^2\mathbb{F}_p+u^{3}\mathbb{F}_{p}+\cdots+u^{k}\mathbb{F}_{p}$ where $u^{k+1}=u$. We illustrate the generator polynomials and investigate the structural properties of these codes via decomposition theorem.
Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ where $p$ is an odd prime, $n$ be a positive integer satisfying ${\rm gcd}(n,p)=1$, and denote $R=\mathbb{F}_{p^m}[u]/\langle u^e\rangle$ where $e\geq 4$ be an even integer. Let $δ,α\in \mathbb{F}_{p^m}^{\times}$. Then the class of $(δ+αu^2)$-constacyclic codes over $R$ is a significant subclass of constacyclic codes over $R$ of Type 2. For any integer $k\geq 1$, an explicit representation and a complete description for all distinct $(δ+αu^2)$-constacyclic codes over $R$ of length $np^k$ and their dual codes are given. Moreover, for…
Let $q$ be a prime power and let $\mathcal{R}=\mathbb{F}_{q}[u_1,u_2, \cdots, u_k]/\langle f_i(u_i),u_iu_j-u_ju_i\rangle$ be a finite non-chain ring, where $f_i(u_i), 1\leq i \leq k$ are polynomials, not all linear, which split into distinct linear factors over $\mathbb{F}_{q}$. We characterize constacyclic codes over the ring $\mathcal{R}$ and study quantum codes from these. As an application, some new and better quantum codes, as compared to the best known codes, are obtained. We also prove that the choice of the polynomials $f_i(u_i),$ $1 \leq i \leq k$ is irrelevant while constructing quan…
In this paper, we study a family of constacyclic BCH codes over $\mathbb{F}_{q^2}$ of length $n=\frac{q^{2m}-1}{q+1}$, where $q$ is a prime power, and $m\geq2$ an even integer. The maximum design distance of narrow-sense Hermitian dual-containing constacyclic BCH codes of length $n$ is determined. Furthermore, the exact dimension of the constacyclic BCH codes with given design distance is computed. As a consequence, we are able to derive the parameters of quantum codes as a function of their design parameters of the associated constacyclic BCH codes. This improves the result by Yuan et al. (De…
