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\([[9,3,3]]\) quadric code[1,2]

Description

Nine-qubit pure Hermitian qubit code constructed from the \([9,3,6]_4\) Hermitian self-orthogonal code. The code can be constructed from the elliptic quadric in \(PG(5, 2)\) [2]. It is the only pure Hermitian code with its parameters [3; ID 170235] and is the highest-distance qubit stabilizer code for its \(n\) and \(k\).

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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
\([[9,3,3]]\) quadric code

References

[1]
I. Bouyukliev and P. R. J. Östergard, “Classification of Self-Orthogonal Codes over \boldmath\(\F_3\) and \boldmath\(\F_4\)”, SIAM Journal on Discrete Mathematics 19, 363 (2005) DOI
[2]
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
[3]
A. Cross and D. Vandeth, “Small Binary Stabilizer Subsystem Codes”, (2025) arXiv:2501.17447
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Zoo Code ID: stab_9_3_3

Cite as:
“\([[9,3,3]]\) quadric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/stab_9_3_3
BibTeX:
@incollection{eczoo_stab_9_3_3, title={\([[9,3,3]]\) quadric code}, booktitle={The Error Correction Zoo}, year={2025}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/stab_9_3_3} }
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Permanent link:
https://errorcorrectionzoo.org/c/stab_9_3_3

Cite as:

“\([[9,3,3]]\) quadric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/stab_9_3_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/9/stab_9_3_3.yml.