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:quant-ph/9711047

Quantum self-dual codes and symmetric matrices

Vladimir D. Tonchev

Published Nov 20, 1997• 0 citations
arXivHTMLPDF

Abstract

This paper has been withdrawn since a Gilbert-Varshamov bound for general quantum codes has already appeared in Ekert and Macchiavello, Prys. Rev. Lett. 77, p. 2585, and a Gilbert-Varshamov bound for stabilizer codes connected with orthogonal geometry, or equivalently, with symmetric matrices as in this paper, has been proved by Calredbank, Rains, Shor and Sloane, Phys. Rev. Lett. 78, p. 405. I would like to thank Robert Calderbank for pointing out these references to me.

DOI
10.48550/arxiv.quant-ph/9711047

Related papers

Symmetric matrices and quantum codes

Vladimir D. Tonchev

This paper has been withdrawn since a Gilbert-Varshamov bound for general quantum codes has already appeared in Ekert and Macchiavello, Prys. Rev. Lett. 77, p. 2585, and a Gilbert-Varshamov bound for stabilizer codes connected with orthogonal geometry, or equivalently, with symmetric matrices as in this paper, has been proved by Calredbank, Rains, Shor and Sloane, Phys. Rev. Lett. 78, p. 405. I would like to thank Robert Calderbank for pointing out these references to me.

arXiv:alg-geom/9711026•Nov 20, 1997•0 citations

QUANTUM SELF-DUAL CODES AND SYMMETRIC MATRICES

Vladimir D. Tonchev

It is proved that a subclass of self-dual quantum-error-correcting codes of length n obtained from binary linear codes of length 2n defined by symmetric matrices, contains codes that meet the Gilbert-Varshamov bound.

•Dec 1, 2000•0 citations

A Varshamov-Gilbert bound for a class of formally self-dual codes and related quantum codes

Vladimir D. Tonchev

It is proved that a class of q-ary (2n,n) formally self-dual codes obtained from symmetric matrices over GF (q), contains codes that meet the Varshamov-Gilbert bound. The codes are self-dual with respect to the symplectic inner product and yield quantum codes encoding one state with n q-ary qubits and having minimum distance proportional to n.

IEEE Transactions on Information Theory•Apr 1, 2002•2 citations

On the List-Decodability of Random Self-Orthogonal Codes

Lingfei Jin, Chaoping Xing, Xiande Zhang

Guruswami et al. showed that the list-decodability of random linear codes is as good as that of general random codes. In this paper, we further strengthen the result by showing that the list-decodability of random Euclidean self-orthogonal codes is as good as that of general random codes as well, i.e., achieves the classical Gilbert-Varshamov bound. In particular, we show that, for any fixed finite field Fq, error fraction δ ∈ (0,1 - 1/q) satisfying 1 - Hq(δ) ≤ 1/2, and small ε > 0, with high probability a random Euclidean self-orthogonal code over Fqof rate 1 - Hq(δ) - ε is (δ, O(1/ε))-list-decodable. This generalizes the result of linear codes to Euclidean self-orthogonal codes. In addition, we extend the result to list decoding symplectic dual-containing codes by showing that the list-decodability of random symplectic dual-containing codes achieves the quantum Gilbert-Varshamov bound as well. This implies that list-decodability of quantum stabilizer codes can achieve the quantum Gilbert-Varshamov bound. The counting argument on self-orthogonal codes is an important ingredient to prove our result.

arXiv:1611.06673•IEEE Transactions on Information Theory•Oct 3, 2014•4 citations

Two Gilbert–Varshamov-type existential bounds for asymmetric quantum error-correcting codes

Ryutaroh Matsumoto

In this note, we report two versions of Gilbert-Varshamov-type existential bounds for asymmetric quantum error-correcting codes.

arXiv:1705.04087•Quantum Information Processing•Oct 23, 2017•11 citations

A Finite Gilbert–Varshamov Bound for Pure Stabilizer Quantum Codes

Kai Feng, Zhi Ma

A finite Gilbert-Varshamov (GV) bound for pure stabilizer (binary and nonbinary) quantum error correcting codes is presented in analogy to the GV bound for classical codes by using several enumerative results in finite unitary geometry. From this quantum GV bound we obtain several new binary quantum codes in a nonconstructive way having better parameters than the known codes.

IEEE Transactions on Information Theory•Nov 30, 2004•110 citations
This paper has been withdrawn since a Gilbert-Varshamov bound for general quantum codes has already appeared in Ekert and Macchiavello, Prys. Rev. Lett. 77, p. 2585, and a Gilbert-Varshamov bound for stabilizer codes connected with orthogonal geometry, or equivalently, with symmetric matrices as in this paper, has been proved by Calredbank, Rains, Shor and Sloane, Phys. Rev. Lett. 78, p. 405. I would like to thank Robert Calderbank for pointing out these references to me.
Guruswami et al. showed that the list-decodability of random linear codes is as good as that of general random codes. In this paper, we further strengthen the result by showing that the list-decodability of random Euclidean self-orthogonal codes is as good as that of general random codes as well, i.e., achieves the classical Gilbert-Varshamov bound. In particular, we show that, for any fixed finite field Fq, error fraction δ ∈ (0,1 - 1/q) satisfying 1 - Hq(δ) ≤ 1/2, and small ε > 0, with high probability a random Euclidean self-orthogonal code over Fqof rate 1 - Hq(δ) - ε is (δ, O(1/ε))-list-decodable. This generalizes the result of linear codes to Euclidean self-orthogonal codes. In addition, we extend the result to list decoding symplectic dual-containing codes by showing that the list-decodability of random symplectic dual-containing codes achieves the quantum Gilbert-Varshamov bound as well. This implies that list-decodability of quantum stabilizer codes can achieve the quantum Gilbert-Varshamov bound. The counting argument on self-orthogonal codes is an important ingredient to prove our result.