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\([[n,n-2k,4]]\) Quantum cap code[1–5]

Description

A distance-four pure Hermitian qubit code constructed by identifying its underlying Hermitian self-orthogonal \([n,k]_4\) code with a particular projective cap in \(PG(k-1,4)\).

Rate

Quantum cap codes tend to have a high rate and include codes with parameters \([[12,4,4]]\) [1], \([[12,2,4]]\), \([[20,10,4]]\), or \([[29,19,4]]\) [4].

Cousin

  • Projective geometry code— A quantum cap code is a distance-four pure Hermitian qubit code constructed by identifying its underlying Hermitian self-orthogonal \([n,k]_4\) code with a particular projective cap in \(PG(k-1,4)\).

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Hermitian qubit codeStabilizer Hamiltonian-based QECC Quantum
Parents
Hermitian qubit code
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
Quantum cap codes tend to have a high rate and include codes with parameters \([[12,4,4]]\) [1], \([[12,2,4]]\), \([[20,10,4]]\), or \([[29,19,4]]\) [4].
\([[n,n-2k,4]]\) Quantum cap code

References

[1]
J. Bierbrauer, G. Faina, M. Giulietti, S. Marcugini, and F. Pambianco, “The geometry of quantum codes”, Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial 6, 53 (2008) DOI
[2]
V. D. Tonchev, “Quantum codes from caps”, Discrete Mathematics 308, 6368 (2008) DOI
[3]
J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
[4]
D. Bartoli, S. Marcugini, and F. Pambianco, “New quantum caps in PG(4,4)”, (2010) arXiv:0905.1059
[5]
Bierbrauer, Jürgen, D. Bartoli, S. Marcugini, and F. Pambianco. "Geometric constructions of quantum codes." Error-Correcting Codes, Finite Geometries and Cryptography 523 (2010): 149-154.
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Zoo Code ID: quantum_cap

Cite as:
“\([[n,n-2k,4]]\) Quantum cap code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/quantum_cap
BibTeX:
@incollection{eczoo_quantum_cap, title={\([[n,n-2k,4]]\) Quantum cap code}, booktitle={The Error Correction Zoo}, year={2025}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_cap} }
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Permanent link:
https://errorcorrectionzoo.org/c/quantum_cap

Cite as:

“\([[n,n-2k,4]]\) Quantum cap code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/quantum_cap

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/quantum_cap.yml.