Quantum Maximum Distance Separable Codes Research Articles

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 .

Quantum error-correcting codes and their geometries

This is an expository article aiming to introduce the reader to the underlying mathematics and geometry of quantum error correction. Information stored on quantum particles is subject to noise and interference from the environment. Quantum error-correcting codes allow the negation of these effects in order to successfully restore the original quantum information. We briefly describe the necessary quantum-mechanical background to be able to understand how quantum error correction works. We go on to construct quantum codes: firstly qubit stabilizer codes, then qubit non-stabilizer codes, and finally codes with a higher local dimension. We will delve into the geometry of these codes. This allows one to deduce the parameters of the code efficiently, deduce the inequivalence between codes that have the same parameters, and presents a useful tool in deducing the feasibility of certain parameters. We also include sections on quantum maximum distance separable codes and the quantum MacWilliams identities.

On geometrically uniform codes and topological quantum MDS codes

In this paper, we present generalized edge-pairings for the family of hyperbolic tessellations {4λ, 4} with the purpose of obtaining the corresponding discrete group of isometries. These tessellations have greater density packing than the self-dual tessellations {4λ, 4λ}, implying that the associated codes achieve the least error probability or, equivalently, that these codes are optimum codes. We also consider the topological quantum error-correcting codes on surfaces with genus g ≥ 2, as proposed in other papers. From that, we focus on the class of topological quantum maximum distance separable codes, showing that the only such codes are precisely the ones with minimum distance d ≤ 2.

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.

Quantum Codes of Maximal Distance and Highly Entangled Subspaces

We present new bounds on the existence of general quantum maximum distance separable codes (QMDS): the length n of all QMDS codes with local dimension D and distance d≥3 is bounded by n≤D2+d−2. We obtain their weight distribution and present additional bounds that arise from Rains' shadow inequalities. Our main result can be seen as a generalization of bounds that are known for the two special cases of stabilizer QMDS codes and absolutely maximally entangled states, and confirms the quantum MDS conjecture in the special case of distance-three codes. As the existence of QMDS codes is linked to that of highly entangled subspaces (in which every vector has uniform r-body marginals) of maximal dimension, our methods directly carry over to address questions in multipartite entanglement.

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.

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.

New entanglement-assisted MDS quantum codes from constacyclic codes

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.

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 $\frac {q}{2}+1$ .

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.

Application of constacyclic codes to entanglement-assisted quantum maximum distance separable codes

The entanglement-assisted stabilizer formalism overcomes the dual-containing constraint of standard stabilizer formalism for constructing quantum codes. This allows ones to construct entanglement-assisted quantum error-correcting codes (EAQECCs) from arbitrary linear codes by pre-shared entanglement between the sender and the receiver. However, it is not easy to determine the number c of pre-shared entanglement pairs required to construct an EAQECC from arbitrary linear codes. In this paper, let q be a prime power, we aim to construct new q-ary EAQECCs from constacyclic codes. Firstly, we define the decomposition of the defining set of constacyclic codes, which transforms the problem of determining the number c into determining a subset of the defining set of underlying constacyclic codes. Secondly, five families of non-Hermitian dual-containing constacyclic codes are discussed. Hence, many entanglement-assisted quantum maximum distance separable codes with $$c\le 7$$ are constructed from them, including ones with minimum distance $$d\ge q+1$$ . Most of these codes are new, and some of them have better performance than ones obtained 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 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.