Quantum Maximum-distance-separable Research Articles

Near MDS and near quantum MDS codes via orthogonal arrays

Near maximum distance separable (NMDS) codes are closely related to interesting objects in finite geometry and have nice applications in combinatorics and cryptography. But there are many unsolved problems about construction of NMDS codes. In this paper, by using symmetrical orthogonal arrays (OAs), we construct a lot of NMDS, m-MDS and almost extremal NMDS codes. Quantum error-correcting codes (QECCs) play a central role in quantum information processing and can protect quantum information from various quantum noises. We present a general method for constructing QECCs over mixed alphabets through asymmetrical OAs. Since quantum maximum distance separable (QMDS) codes over mixed alphabets with the dimension equal to one have not been found in all the literature so far, the definition of a near QMDS code over mixed alphabets is proposed. By using asymmetrical OAs, we obtain many such codes.

Several classes of Galois self-orthogonal MDS codes and related applications

Let q=ph be a prime power and e be an integer with 0≤e≤h−1. e-Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes (e=0) and Hermitian self-orthogonal codes (e=h2 and h is even). In this paper, we propose two general methods to construct e-Galois self-orthogonal (extended) generalized Reed-Solomon (GRS) codes. As a consequence, eight new classes of e-Galois self-orthogonal (extended) GRS codes with odd q and 2e|h are obtained. Based on the Galois dual of a code, we also study its punctured and shortened codes. As applications, new e′-Galois self-orthogonal maximum distance separable (MDS) codes for all possible e′ satisfying 0≤e′≤h−1, new e-Galois self-orthogonal MDS codes via the shortened codes, and new MDS codes with prescribed dimensional e-Galois hull via the punctured codes are derived. Moreover, some new q-ary quantum MDS codes with length greater than q+1 and minimum distance greater than q2+1 are obtained.

Some quantum MDS codes from GRS codes

ABSTRACT The construction of quantum maximum-distance-separable (MDS) codes has become one of the major goals in quantum coding theory. In this paper, we construct several classes of Hermitian self-orthogonal generalized Reed–Solomon codes. Based on these classical MDS codes, we obtain several new classes of quantum MDS codes with large minimum distance. It turns out that these constructed quantum MDS codes have fewer constraints on the selection of code length, and in some cases, have larger minimum distance than previous literature. Notably, some of the distance parameters of our codes are greater than ( q / 2 ) + 1 .

Some new quantum MDS codes derived from generalized Reed-Solomon codes

<p style='text-indent:20px;'>Quantum maximum-distance-separable (MDS) codes are very important in coding theory. In this paper, we construct three classes of new quantum MDS codes with large minimum distance by classical Hermitian self-orthogonal generalized Reed-Solomon codes. Furthermore, these quantum MDS codes have more flexible lengths and larger minimum distance than those of some known quantum MDS codes.</p>

Quantum Codes from Repeated-Root Cyclic and Negacyclic Codes of Length 4ps Over $\mathbb {F}_{p^{m}}$

Let $\mathbb {F}_{p^{m}}$ be a finite field with pm elements for an odd prime p and a positive integer m. In this paper, we aim to construct non-binary quantum codes from repeated root cyclic and negacyclic codes of length 4ps over $\mathbb {F}_{p^{m}}$ . To achieve this, first we discuss the structure of all maximum distance separable (MDS) cyclic and negacyclic codes of length 4ps over $\mathbb {F}_{p^{m}}$ and their Euclidean and Hermitian duals. Further, we establish a necessary and sufficient condition for these codes to contain their duals. Finally, many new and quantum MDS codes are obtained with the help of CSS and Hermitian constructions from dual containing cyclic and negacyclic codes.

Absolutely maximally entangled states, quantum-maximum-distance-separable codes, and quantum repeaters

We extend the relation between absolutely maximally entangled (AME) states and quantum maximum distance separable (QMDS) codes by constructing whole families of QMDS codes from their parent AME states. We introduce a reduction-friendly form for the generator set of the stabilizer representation of an AME state, from which the stabilizer form for children codes, all QMDS, can be obtained. We then relate this to optimal codes for one-way quantum repeaters, by minimizing the short-term infrastructure cost as well as the long-term running cost of such quantum repeaters. We establish that AME states provide a framework for a class of QMDS codes that can be used in quantum repeaters.

Some New Constructions of Quantum MDS Codes

It is an important task to construct quantum maximum-distance-separable (MDS) codes with good parameters. In the present paper, we provide six new classes of q-ary quantum MDS codes by using generalized Reed-Solomon (GRS) codes and Hermitian construction. The minimum distances of our quantum MDS codes can be larger than q/2+1 Three of these six classes of quantum MDS codes have longer lengths than the ones constructed in [1] and [2], hence some of their results can be easily derived from ours via the propagation rule. Moreover, some known quantum MDS codes of specific lengths can be seen as special cases of ours and the minimum distances of some known quantum MDS codes are also improved as well.

A Different Construction for Some Classes of Quantum MDS Codes

Constructing quantum codes with large minimum distance plays a significant role in quantum computation and communication. Quantum maximum-distance separable (MDS) codes have important place among quantum codes since they are optimal with regard to the maximality of their minimum distances. In this paper, by making use of constacyclic codes over $$F_{q^2}$$ and Hermitian construction for quantum codes, we give different constructions for some classes of quantum MDS codes.

Some new quantum MDS codes from generalized Reed–Solomon codes

Quantum maximum distance separable (MDS) codes form a significant class of quantum codes. In this paper, by using Hermitian self-orthogonal generalized Reed–Solomon codes, we construct two new classes of q-ary quantum MDS codes, which have minimum distance greater than q2. Most of these quantum MDS codes are new in the sense that their parameters are not covered by the codes available in the literature.

New quantum MDS codes with large minimum distance and short length from generalized Reed–Solomon codes

Quantum maximum-distance-separable (MDS) codes are an important class of quantum codes. In this paper we mainly use classical Hermitian self-orthogonal generalized Reed–Solomon codes to construct three classes of new quantum MDS codes. Further, these quantum MDS codes have large minimum distance and short length.

Applications of Constacyclic Codes to Some New Entanglement-Assisted Quantum MDS Codes

Generally, it is not easy to construct quantum maximal-distance-separable (MDS) codes with the minimum distance greater than $\frac {q}{2}+1$ . The minimum distance of quantum MDS codes can achieve $\frac {q}{2}+1$ or exceed $\frac {q}{2}+1$ by adopting pre-shared entanglement. In this work, some new families of entanglement-assisted quantum MDS codes that satisfy the quantum Singleton bound are constructed and the number of maximally entangled states required is determined to make the minimum distance of some constructed codes achieve $\frac {q}{2}+1$ or exceed $\frac {q}{2}+1$ by utilizing the decomposition of the defining set and $q^{2}$ -cyclotomic cosets of constacyclic codes with length $\frac {q^{2}+1}{\gamma }$ , where $\gamma =t^{2}+1$ , $t$ is a power of 2 and $q=t^{e}>4$ with $e\equiv 1~\textrm {mod}~4$ or $e\equiv 3~\textrm {mod}~4$ . Moreover, the parameters of these codes constructed in this paper are more general relative to the ones in the literature and the minimum distance of some codes constructed in this paper is larger than $\frac {q}{2}+1$ .

Some New Classes of Entanglement-Assisted Quantum MDS Codes Derived From Constacyclic Codes

Although quantum maximal-distance-separable (MDS) codes that satisfy the quantum singleton bound have become an important research topic in the quantum coding theory, it is not an easy task to search for quantum MDS codes with the minimum distance that is larger than (q/2) + 1. The pre-shared entanglement between the sender and the receiver can improve the minimum distance of quantum MDS codes such that the minimum distance of some constructed codes achieves (q/2) + 1 or exceeds (q/2) + 1. Meanwhile, how to determine the required number of maximally entangled states to make the minimum distance of quantum MDS codes larger than (q/2) + 1 is an interesting problem in the quantum coding theory. In this paper, we utilize the decomposition of the defining set and q 2 -cyclotomic cosets of constacyclic codes with the form q = αm + t or q = αm + α - t and n = (q 2 + 1/α) to construct some new families of entanglement-assisted quantum MDS codes that satisfy the entanglement-assisted quantum singleton bound, where q is an odd prime power and m is a positive integer, while both α and t are positive integers such that α = t 2 + 1. The parameters of these codes constructed in this paper are more general compared with the ones in the literature. Moreover, the minimum distance of some codes in this paper is larger than (q/2) + 1 or q + 1.

Quantum MDS codes from BCH constacyclic codes

One central theme in quantum error correction is to construct quantum codes that have large minimum distances. It has been a great challenge to construct new quantum maximum-distance-separable (MDS) codes. Recently, some quantum MDS codes have been constructed from constacyclic codes. Under these constructions, one of the most important problems is to ensure these constacyclic codes are Hermitian dual-containing. This paper presents a method for determining the maximal designed distance of $$[[n, k, d]]_q$$ quantum MDS codes from constacyclic codes with fixed n and q. From the method, we can get not only those known quantum MDS codes from constacyclic codes but also a new class of quantum MDS code from Hermitian dual-containing MDS constacyclic code.

On the Construction of Quantum MDS Codes

Quantum maximal-distance-separable (MDS) codes form an important class of quantum codes. The paper is devoted to the construction of new quantum MDS codes based on classical constacyclic codes and cyclic codes over $\mathbb {F}_{q^{2}}$ . We construct a family of new quantum MDS codes with length $n=\frac {q^{2}+ 1}{a}$ , which is a generalization of results in Zhang and Ge (IEEE Trans. Inf. Theory 61(9):5224–5228, 2015), Chen et al. (IEEE. Trans. Inf. Theory 61:1474–1484, 2015). Most of these quantum MDS codes are new in the sense that their parameters are not covered by the quantum codes available in the literature.

Two new classes of quantum MDS codes

Let p be a prime and let q be a power of p. In this paper, by using generalized Reed–Solomon (GRS for short) codes and extended GRS codes, we construct two new classes of quantum maximum-distance-separable (MDS) codes with parameters[[tq,tq−2d+2,d]]q for any 1≤t≤q,2≤d≤⌊tq+q−1q+1⌋+1, and[[t(q+1)+2,t(q+1)−2d+4,d]]q for any 1≤t≤q−1,2≤d≤t+2 with (p,t,d)≠(2,q−1,q). Our quantum MDS codes have flexible parameters, and have minimum distances larger than q2+1 when t>q2. Furthermore, it turns out that our constructions generalize and improve some previous results.

Some families of optimal quantum codes derived from constacyclic codes

ABSTRACTQuantum maximum-distance-separable (MDS) codes form an important family of quantum codes. Quantum MDS codes are optimal, which can attain the quantum Singleton bound. In this paper, using some classes of constacyclic codes and Hermitian constructions, we propose the constructions of optimal quantum convolutional codes and optimal asymmetric quantum codes in the sense of the quantum Singleton bound. Compared with the codes obtained in the literature, the parameters of the constructed quantum codes are new as well as the distance of optimal asymmetric quantum codes constructed in this paper is larger.

Some quantum MDS codes with large minimum distance from generalized Reed-Solomon codes

Quantum maximum-distance-separable (MDS) codes are a significant class of quantum codes. In this paper, we mainly utilize classical Hermitian self-orthogonal generalized Reed-Solomon codes to construct five new classes of quantum MDS codes with large minimum distance.

Some Families of Asymmetric Quantum MDS Codes Constructed from Constacyclic Codes

Quantum maximal-distance-separable (MDS) codes that satisfy quantum Singleton bound with different lengths have been constructed by some researchers. In this paper, seven families of asymmetric quantum MDS codes are constructed by using constacyclic codes. We weaken the case of Hermitian-dual containing codes that can be applied to construct asymmetric quantum MDS codes with parameters $[[n,k,d_{z}/d_{x}]]_{q^{2}}$ . These quantum codes are able to correct quantum errors with greater asymmetry. Moreover, these quantum codes constructed in this paper are different from the ones in the literature.

Construction of some new quantum MDS codes

Quantum maximum-distance-separable (MDS) codes are an important class of quantum codes. In this paper, we mainly apply a new method of classical Hermitian self-orthogonal codes to construct three classes of new quantum MDS codes, and these quantum MDS codes provide large minimum distance.

Construction of new quantum MDS codes derived from constacyclic codes

Obtaining quantum maximum distance separable (MDS) codes from dual containing classical constacyclic codes using Hermitian construction have paved a path to undertake the challenges related to such constructions. Using the same technique, some new parameters of quantum MDS codes have been constructed here. One set of parameters obtained in this paper has achieved much larger distance than work done earlier. The remaining constructed parameters of quantum MDS codes have large minimum distance and were not explored yet.