Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1502.05267 · IEEE International Symposium on Information Theory (ISIT)

Quantum MDS codes over small fields

Markus Grassl, Martin Roetteler

Published Jun 1, 2015• 79 citations
arXivHTMLPDF

Abstract

We consider quantum MDS (QMDS) codes for quantum systems of dimension q with lengths up to q2+ 2 and minimum distances up to q + 1. We show how starting from QMDS codes of length q2+ 1 based on cyclic and constacyclic codes, new QMDS codes can be obtained by shortening. We provide numerical evidence for our conjecture that almost all admissible lengths, from a lower bound n0(q, d) on, are achievable by shortening. Some additional codes that fill gaps in the list of achievable lengths are presented as well along with a construction of a family of QMDS codes of length q2+2, where q = 2m, that appears to be new.

DOI
10.1109/isit.2015.7282626

Related papers

On quantum MDS codes

Martin Rötteler, Markus Grassl, T. Beth

We construct maximum distance separable quantum error-correcting codes. The codes are defined over q-dimensional quantum systems, where q is any prime power. The construction yields quantum MDS codes of length up to q+1 for all possible dimensions and some quantum MDS codes of length up to q2+1. In particular, those families contain codes [[6,2,3]]pand [[7,3,3]]pfor pges3 (cf. [K. Feng (2002)])

IEEE International Symposium on Information Theory (ISIT)•Dec 23, 2004•55 citations

Quantum MDS and Synchronizable Codes From Cyclic and Negacyclic Codes of Length 2ps Over F pm

Hai Quang Dinh, Bac Trong Nguyen, Woraphon Yamaka

Let p be an odd prime, and Fp(m)is the finite field of pmelements. In this paper, all maximum distance separable (briefly, MDS) cyclic and negacyclic codes of length 2psover Fp(m)are established. As an application, all quantum MDS (briefly, qMDS) codes are constructed from cyclic and negacyclic codes of length 2psover finite fields using the Calderbank Shor-Steane (briefly, CSS) and Hermitian constructions. These codes are new in the sense that their parameters are different from all the previous constructions. Furthermore, quantum synchronizable codes (briefly, QSCs) are obtained from cyclic codes of length 2psover Fp(m). To enrich the variety of available QSCs, many new QSCs are constructed to illustrate our results. Among them, there are QSCs codes with shorter lengths and much larger minimum distances than known primitive narrow-sense Bose-Chaudhuri-Hocquenghem (briefly, BCH) codes.

IEEE Access•Jan 1, 2020•16 citations

Quantum MDS Codes with length $n\equiv 0,1($mod$\,\frac{q\pm1}{2})$

Ruhao Wan

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 $n$ of most of the quantum MDS codes we constructed satisfies $n\equiv 0,1($mod$\,\frac{q\pm1}{2})$, 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 $q/2+1$.

arXiv:2310.00214•Sep 30, 2023•0 citations

Some New Classes of Quantum MDS Codes From Constacyclic Codes

Tao Zhang, Gennian Ge

Quantum maximum-distance-separable (MDS) codes form an important family of quantum codes. In this paper, using Hermitian construction and classical constacyclic codes, we construct six classes of quantum MDS codes. Two of these six classes of quantum MDS codes have larger minimum distance than the ones available in the literature. Most of these quantum MDS codes are new in the sense that their parameters are not covered by the codes available in the literature.

IEEE Transactions on Information Theory•Jul 21, 2015•66 citations

A Construction of New Quantum MDS Codes

Lingfei Jin, Chaoping Xing

It has been a great challenge to construct new quantum maximum-distance-separable (MDS) codes. In particular, it is very hard to construct the quantum MDS codes with relatively large minimum distance. So far, except for some sparse lengths, all known q-ary quantum MDS codes have minimum distance ≤q/2 + 1. In this paper, we provide a construction of the quantum MDS codes with minimum distance >q/2 + 1. In particular, we show the existence of the q-ary quantum MDS codes with length n = q2+ 1 and minimum distance d for any d q + 1 (this result extends those given in the works of Guardia (2011), Jin et al. (2010), and Kai an Zhu (2012)); and with length (q2+ 2)/3 and minimum distance d for any d (2q+2)/3 if 3|(q + 1). Our method is through Hermitian selforthogonal codes. The main idea of constructing the Hermitian self-orthogonal codes is based on the solvability in Fqof a system of homogenous equations over Fq2.

arXiv:1311.3009•IEEE Transactions on Information Theory•Jan 31, 2014•108 citations

ON OPTIMAL QUANTUM CODES

Markus Grassl, Thomas Beth, Martin Rötteler

We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters 〚n, n - 2d + 2, d〛q exist for all 3≤n≤q and 1≤d≤n/2+1. We also present quantum MDS codes with parameters 〚q2, q2-2d+2, d〛q for 1≤d≤q which additionally give rise to shortened codes 〚q2-s, q2-2d+2-s, d〛q for some s.

arXiv:quant-ph/0312164•International Journal of Quantum Information•Mar 1, 2004•285 citations
We consider quantum MDS (QMDS) codes for quantum systems of dimension q with lengths up to q2+ 2 and minimum distances up to q + 1. We show how starting from QMDS codes of length q2+ 1 based on cyclic and constacyclic codes, new QMDS codes can be obtained by shortening. We provide numerical evidence for our conjecture that almost all admissible lengths, from a lower bound n0(q, d) on, are achievable by shortening. Some additional codes that fill gaps in the list of achievable lengths are presented as well along with a construction of a family of QMDS codes of length q2+2, where q = 2m, that appears to be new.
Let p be an odd prime, and Fp(m)is the finite field of pmelements. In this paper, all maximum distance separable (briefly, MDS) cyclic and negacyclic codes of length 2psover Fp(m)are established. As an application, all quantum MDS (briefly, qMDS) codes are constructed from cyclic and negacyclic codes of length 2psover finite fields using the Calderbank Shor-Steane (briefly, CSS) and Hermitian constructions. These codes are new in the sense that their parameters are different from all the previous constructions. Furthermore, quantum synchronizable codes (briefly, QSCs) are obtained from cyclic codes of length 2psover Fp(m). To enrich the variety of available QSCs, many new QSCs are constructed to illustrate our results. Among them, there are QSCs codes with shorter lengths and much larger minimum distances than known primitive narrow-sense Bose-Chaudhuri-Hocquenghem (briefly, BCH) codes.
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.
It has been a great challenge to construct new quantum maximum-distance-separable (MDS) codes. In particular, it is very hard to construct the quantum MDS codes with relatively large minimum distance. So far, except for some sparse lengths, all known q-ary quantum MDS codes have minimum distance ≤q/2 + 1. In this paper, we provide a construction of the quantum MDS codes with minimum distance >q/2 + 1. In particular, we show the existence of the q-ary quantum MDS codes with length n = q2+ 1 and minimum distance d for any d q + 1 (this result extends those given in the works of Guardia (2011), Jin et al. (2010), and Kai an Zhu (2012)); and with length (q2+ 2)/3 and minimum distance d for any d (2q+2)/3 if 3|(q + 1). Our method is through Hermitian selforthogonal codes. The main idea of constructing the Hermitian self-orthogonal codes is based on the solvability in Fqof a system of homogenous equations over Fq2.
We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters 〚n, n - 2d + 2, d〛q exist for all 3≤n≤q and 1≤d≤n/2+1. We also present quantum MDS codes with parameters 〚q2, q2-2d+2, d〛q for 1≤d≤q which additionally give rise to shortened codes 〚q2-s, q2-2d+2-s, d〛q for some s.