Constructions of q-ary entanglement-assisted quantum MDS codes with minimum distance greater than q+1

he entanglement-assisted stabilizer formalism provides a useful framework for constructing quantum error-correcting codes (QECC), which can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this paper, we construct five classes of entanglement-assisted quantum MDS (EAQMDS) codes based on classical MDS codes by exploiting one or more pre-shared maximally entangled states. We show that these EAQMDS codes have much larger minimum distance than the standard quantum MDS (QMDS) codes of the same length, and three classes of these EAQMDS codes consume only one pair of maximally entangled states.

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New entanglement-assisted quantum MDS codes

International Journal of Quantum Information ◽

10.1142/s0219749921500167 ◽

2021 ◽

pp. 2150016

Author(s):

Binbin Pang ◽

Shixin Zhu ◽

Liqi Wang

Keyword(s):

Coding Theory ◽

Minimum Distance ◽

Linear Codes ◽

Error Correcting Codes ◽

Mds Codes ◽

Maximum Distance Separable ◽

Number Formula ◽

Minimum Distances ◽

Quantum Error ◽

Quantum Mds Codes

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.

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Constructing new q-ary quantum MDS codes with distances bigger than q/2 from generator matrices

Quantum Information and Computation ◽

10.26421/qic18.3-4-3 ◽

2018 ◽

Vol 18 (3&4) ◽

pp. 223-230

Author(s):

Xianmang He

Keyword(s):

Information Theory ◽

Quantum Information ◽

Minimum Distance ◽

Quantum Information Theory ◽

Error Correcting Codes ◽

Mds Codes ◽

Quantum Error ◽

Generator Matrices ◽

Quantum Mds Codes

The construction of quantum error-correcting codes has been an active field of quantum information theory since the publication of \cite{Shor1995Scheme,Steane1998Enlargement,Laflamme1996Perfect}. It is becoming more and more difficult to construct some new quantum MDS codes with large minimum distance. In this paper, based on the approach developed in the paper \cite{NewHeMDS2016}, we construct several new classes of quantum MDS codes. The quantum MDS codes exhibited here have not been constructed before and the distance parameters are bigger than q/2.

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ON OPTIMAL QUANTUM CODES

International Journal of Quantum Information ◽

10.1142/s0219749904000079 ◽

2004 ◽

Vol 02 (01) ◽

pp. 55-64 ◽

Cited By ~ 117

Author(s):

MARKUS GRASSL ◽

THOMAS BETH ◽

MARTIN RÖTTELER

Keyword(s):

Minimum Distance ◽

Prime Power ◽

Quantum Systems ◽

Error Correcting Codes ◽

Mds Codes ◽

Quantum Codes ◽

Maximum Distance Separable ◽

Shortened Codes ◽

Quantum Error ◽

Quantum Mds Codes

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.

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New constructions of entanglement-assisted quantum MDS codes from negacyclic codes

International Journal of Quantum Information ◽

10.1142/s0219749919500229 ◽

2019 ◽

Vol 17 (03) ◽

pp. 1950022 ◽

Cited By ~ 6

Author(s):

Ruihu Li ◽

Guanmin Guo ◽

Hao Song ◽

Yang Liu

Keyword(s):

Minimum Distance ◽

Linear Codes ◽

Error Correcting Codes ◽

Optimal Number ◽

Mds Codes ◽

Quantum Codes ◽

General Statement ◽

Negacyclic Codes ◽

Maximum Distance Separable ◽

Quantum Error

When constructing quantum codes under the entanglement-assisted (EA) stabilizer formalism, one can ignore the limitation of dual-containing condition. This allows us to construct EA quantum error-correcting codes (QECCs) from any classical linear codes. The main contribution of this manuscript is to make a general statement for determining the optimal number of pre-shared qubits instead of presenting only specific cases. Let [Formula: see text] and [Formula: see text], where [Formula: see text] is an odd prime power, [Formula: see text] and [Formula: see text]. By deeply investigating the decomposition of the defining set of negacyclic codes, we generalize the number of pre-shared entanglement pairs of Construction (1) in Lu et al. [Quantom Inf. Process. 17 (2018) 69] from [Formula: see text] to arbitrary even numbers less than or equal to [Formula: see text]. Consequently, a series of EA quantum maximum distance separable (EAQMDS) codes can be produced. The absolute majority of them are new and the minimum distance can be up to [Formula: see text]. Moreover, this method can be applied to construct many other families of EAQECCs with good parameters, especially large minimum distance.

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Optimal quantum error correcting codes from absolutely maximally entangled states

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8121/aaa151 ◽

2018 ◽

Vol 51 (7) ◽

pp. 075301 ◽

Cited By ~ 13

Author(s):

Zahra Raissi ◽

Christian Gogolin ◽

Arnau Riera ◽

Antonio Acín

Keyword(s):

Error Correcting Codes ◽

Entangled States ◽

Maximally Entangled States ◽

Quantum Error

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Quantum error-correction codes and absolutely maximally entangled states

Physical Review A ◽

10.1103/physreva.101.042305 ◽

2020 ◽

Vol 101 (4) ◽

Cited By ~ 1

Author(s):

Paweł Mazurek ◽

Máté Farkas ◽

Andrzej Grudka ◽

Michał Horodecki ◽

Michał Studziński

Keyword(s):

Error Correction ◽

Quantum Error Correction ◽

Entangled States ◽

Error Correction Codes ◽

Maximally Entangled States ◽

Quantum Error Correction Codes ◽

Quantum Error

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Entanglement-assisted quantum codes constructed from primitive quaternary BCH codes

International Journal of Quantum Information ◽

10.1142/s0219749914500154 ◽

2014 ◽

Vol 12 (03) ◽

pp. 1450015 ◽

Cited By ~ 22

Author(s):

Liang-Dong Lü ◽

Ruihu Li

Keyword(s):

Linear Codes ◽

Careful Analysis ◽

Galois Field ◽

Error Correcting Codes ◽

Optimal Number ◽

Bch Codes ◽

Design Parameters ◽

Bch Code ◽

Cyclotomic Cosets ◽

Quantum Error

The entanglement-assisted (EA) formalism generalizes the standard stabilizer formalism. All quaternary linear codes can be transformed into entanglement-assisted quantum error correcting codes (EAQECCs) under this formalism. In this work, we discuss construction of EAQECCs from Hermitian non-dual containing primitive Bose–Chaudhuri–Hocquenghem (BCH) codes over the Galois field GF(4). By a careful analysis of the cyclotomic cosets contained in the defining set of a given BCH code, we can determine the optimal number of ebits that needed for constructing EAQECC from this BCH code, rather than calculate the optimal number of ebits from its parity check matrix, and derive a formula for the dimension of this BCH code. These results make it possible to specify parameters of the obtained EAQECCs in terms of the design parameters of BCH codes.

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