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 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 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_…
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.
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…
Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$, $λ$ and $k$ be integers satisfying $λ,k\geq 2$ and denote $R=\mathbb{F}_{2^m}[u]/\langle u^{2λ}\rangle$. Let $δ,α\in \mathbb{F}_{2^m}^{\times}$. For any odd positive integer $n$, we give an explicit representation and enumeration for all distinct $(δ+αu^2)$-constacyclic codes over $R$ of length $2^kn$, and provide a clear formula to count the number of all these codes. As a corollary, we conclude that every $(δ+αu^2)$-constacyclic code over $R$ of length $2^kn$ is an ideal generated by at most $2$ polynomials in the residue class r…
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…
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…
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…
Let $\mathbb{F}_q$ be a finite field of order $q$ and integer $n\ge 1$. Let $\ell$ be a prime such that $\ell^k|(q-1)$ for some integer $k\ge 1$ and $μ$ be an element of order $\ell^k$ in $\mathbb{F}_q$. In this paper, we determine the weight distributions of all irreducible $μ$-constacyclic codes of length $\ell^n$ over $\mathbb{F}_q$. Explicit expressions for the generator polynomials and codewords of these codes are also obtained.
Locally repairable codes (LRCs) have emerged as an important coding scheme in distributed storage systems (DSSs) with relatively low repair cost by accessing fewer non-failure nodes. Theoretical bounds and optimal constructions of LRCs have been widely investigated. Optimal LRCs via cyclic and constacyclic codes provide significant benefit of elegant algebraic structure and efficient encoding procedure. In this paper, we continue to consider the constructions of optimal LRCs via cyclic and constacyclic codes with long code length. Specifically, we first obtain two classes of $q$-ary cyclic Sin…
$(1+2u)$-constacyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$
设$R=\mathbb{Z}_4+u\mathbb{Z}_4,$，其中$\mathbb{Z}_4$表示模$4$和$u^2=0$的整数环。本文引入了$R^n$到$\mathbb{Z}_{4}^{2n}.$的一个新的Gray映射，并借助于$R$上的循环码研究了$R$上的奇数长$(1+2u)$ -常循环码.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.进一步得到了$\mathbb{Z}_4$上的若干线性码作为$R$上的$(1+2u)$ -常循环码的映象.
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。
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…
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…
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 $p$ be an odd prime, and let $m$ be a positive integer satisfying $p^m \equiv 3~(\text{mod }4).$ Let $\mathbb{F}_{p^m}$ be the finite field with $p^m$ elements, and let $R=\mathbb{F}_{p^m}[u]/\left<u^2\right>$ be the finite commutative chain ring with unity. In this paper, we determine all constacyclic codes of length $4p^s$ over $R$ and their dual codes, where $s$ is a positive integer. We also determine their sizes and list some isodual constacyclic codes of length $4p^s$ over $R.$
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.
For prime $p$, $GR(p^a,m)$ represents the Galois ring of order $p^{am}$ and characterise $p$, where $a$ is any positive integer. In this article, we study the Type (1) $λ$-constacyclic codes of length $4p^s$ over the ring $GR(p^a,m)$, where $λ=ξ_0+pξ_1+p^2z$, $ξ_0,ξ_1\in T(p,m)$ are nonzero elements and $z\in GR(p^a,m)$. In first case, when $λ$ is a square, we show that any ideal of $\mathcal{R}_p(a,m,λ)=\frac{GR(p^a,m)[x]}{\langle x^{4p^s}-λ\rangle}$ is the direct sum of the ideals of $\frac{GR(p^a,m)[x]}{\langle x^{2p^s}-δ\rangle}$ and $\frac{GR(p^a,m)[x]}{\langle x^{2p^s}+δ\rangle}$. In sec…
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…
In this paper, we determine depth spectra of all repeated-root $(α+γβ)$-constacyclic codes of arbitrary lengths over a finite commutative chain ring $\mathcal{R},$ where $α$ is a non-zero element of the Teichmüller set of $\mathcal{R},$ $γ$ is a generator of unique maximal ideal of $\mathcal{R}$ and $β$ is a unit in $\mathcal{R}.$ We also illustrate our results with some examples.
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…
Symbol-pair codes are block codes with symbol-pair metrics designed to protect against pair-errors that may occur in high-density data storage systems. MDS symbol-pair codes are optimal in the sense that it can attain the highest pair-error correctability within the same code length and code size. Constructing MDS symbol-pair codes is one of the main topics in symbol-pair codes. In this paper, we characterize the symbol-pair distances of some constacyclic codes of arbitrary lengths over finite fields and a class of finite chain rings. Using the characterization of symbol-pair distance, we pres…
Constacyclic symbol-pair codes: lower bounds and optimal constructions
Cassuto和Blaum（2010）引入的符号对代码旨在防止符号对读取通道中的对错误。最小对距离越大，代码可以纠正的对错误就越多。MDS码元对码是最佳的意义上说，对距离不能提高给定的长度和代码的大小。本文的贡献是双重的。首先，我们给出了常循环码的最小对距离的三个下界，其中前两个推广了Cassuto和Blaum（2011）和Kai {\it et al.}（2015年）。第三个模型给出了重根循环码最小码对距离的一个下界。其次，我们利用重根循环码得到了新的最小码对距离为7和8的MDS码对。
We introduce new sufficient conditions for permutation and monomial equivalence of linear cyclic codes over various finite fields. We recall that monomial equivalence and isometric equivalence are the same relation for linear codes over finite fields. A necessary and sufficient condition for the monomial equivalence of linear cyclic codes through a shift map on their defining set is also given. Moreover, we provide new algebraic criteria for the monomial equivalence of constacyclic codes over $\mathbb{F}_4$. Finally, we prove that if $\gcd(3n,ϕ(3n))=1$, then all permutation equivalent constacy…
