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IEEE Access

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

Jian‐Zhang Chen, Youqin Chen, Dong Yu, Chunhui Feng, Yuanyuan Huang, Riqing Chen

Published Jan 1, 2019• 4 citations
Publisher

Abstract

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

DOI
10.1109/access.2019.2941416

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Generally, it is not easy to construct quantum maximal-distance-separable (MDS) codes with the minimum distance greater thanq2+1\frac {q}{2}+12q​+1. The minimum distance of quantum MDS codes can achieveq2+1\frac {q}{2}+12q​+1or exceedq2+1\frac {q}{2}+12q​+1by 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 achieveq2+1\frac {q}{2}+12q​+1or exceedq2+1\frac {q}{2}+12q​+1by utilizing the decomposition of the defining set andq2q^{2}q2-cyclotomic cosets of constacyclic codes with lengthq2+1γ\frac {q^{2}+1}{\gamma }γq2+1​, whereγ=t2+1\gamma =t^{2}+1γ=t2+1,tttis a power of 2 andq=te>4q=t^{e}>4q=te>4withe≡1 mod 4e\equiv 1~\textrm {mod}~4e≡1 mod 4ore≡3 mod 4e\equiv 3~\textrm {mod}~4e≡3 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 thanq2+1\frac {q}{2}+12q​+1.
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 q2-cyclotomic cosets of constacyclic codes with the form q = αm + t or q = αm + α - t and n = (q2+ 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 = t2+ 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.
Entanglement-assisted quantum error-correcting codes can be seen as a new-type of quantum error-correcting codes and can be constructed from arbitrary linear codes which should not satisfy the dual-containing condition by utilizing shared entangled states between the sender and the receiver in advance. In this paper, we construct several new classes of entanglement-assisted quantum maximum-distance-separable codes from constacyclic codes and cyclic codes by exploiting small pre-shared entangled states, respectively. These codes are new in the sense that they are not covered by the codes available in the literature.
Quantum error correction is fundamentally important for quantum information processing and computation. Quantum error correction codes have been studied and constructed since the pioneering papers of Shor and Steane. Optimal (called MDS) qqq-qubit quantum codes attaining the quantum Singleton bound were constructed for very restricted lengths n≤q2+1n \leq q^2+1n≤q2+1. Entanglement-assisted quantum error correction (EAQEC) code was proposed to use the pre-shared maximally entangled state for the enhancing of error correction capability. Recently there have been a lot of constructions of MDS EAQEC codes attaining the quantum Singleton bound for very restricted lengths. In this paper we construct such MDS EAQEC [[n,k,d,c]]q[[n, k, d, c]]_q[[n,k,d,c]]q​ codes for arbitrary nnn satisfying n≤q2+1n \leq q^2+1n≤q2+1 and arbitrary distance d≤n+22d\leq \frac{n+2}{2}d≤2n+2​. It is proved that for any given length nnn satisfying O(q2)=n≤q2+1O(q^2)=n \leq q^2+1O(q2)=n≤q2+1 and any given distance ddd satisfying $ O(q^2)=d \leq \frac{n+2}{2}$, there exist at least O(q2)O(q^2)O(q2) MDS EAQEC [[n,k,d,c]]q[[n, k, d, c]]_q[[n,k,d,c]]q​ codes with different ccc parameters. Our results show that there are much more MDS entanglement-assisted quantum codes than MDS quantum codes without consumption of the maximally entangled state. This is natural from the physical point of view. Our method can also be applied to construct MDS entanglement-assisted quantum codes from the generalized MDS twisted Reed-Solomon codes.
Quantum maximum-distance-separable (MDS) codes form an important class of quantum codes. To get q-ary quantum MDS codes, one of the effective ways is to find linear MDS codes C over Fq2satisfying C⊥H⊆ C, where C⊥Hdenotes the Hermitian dual code of C. For a linear code C of length n over Fq2, we say that C is a dual-containing code if C⊥H⊆ C and C≠ Fq2n. Several classes of new quantum MDS codes with relatively large minimum distance have been produced through dual-containing constacyclic MDS codes. These works motivate us to make a careful study on the existence conditions for dual-containing constacyclic codes. We obtain necessary and sufficient conditions for the existence of dual-containing constacyclic codes. Four classes of dual-containing constacyclic MDS codes are constructed and their parameters are computed. Consequently, the quantum MDS codes are derived from these parameters. The quantum MDS codes exhibited here have minimum distance bigger than the ones available in the literature.
Entanglement-assisted quantum error-correcting codes (EAQECCs) can be obtained from arbitrary classical linear codes based on the entanglement-assisted stabilizer formalism, which greatly promoted the development of quantum coding theory. In this paper, we construct several families of [Formula: see text]-ary entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes of lengths [Formula: see text] with flexible parameters as to the minimum distance [Formula: see text] and the number [Formula: see text] of maximally entangled states. Most of the obtained EAQMDS codes have larger minimum distances than the codes available in the literature.
Quantum MDS Codes with length n≡0,1(n\equiv 0,1(n≡0,1(mod q±12)\,\frac{q\pm1}{2})2q±1​)
An important family of quantum codes is the quantum maximum-distance-separable (MDS) codes. In this paper, we construct some new classes of quantum MDS codes by generalized Reed-Solomon (GRS) codes and Hermitian construction. In addition, the length nnn of most of the quantum MDS codes we constructed satisfies n≡0,1(n\equiv 0,1(n≡0,1(mod q±12)\,\frac{q\pm1}{2})2q±1​), which is different from previously known code lengths. At the same time, the quantum MDS codes we construct have large minimum distances that are greater than q/2+1q/2+1q/2+1.