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\([[30,8,3]]\) Bring code[1]

Alternative names: Small stellated dodecahedron code.

Description

A \([[30,8,3]]\) hyperbolic surface code on a quotient of the \(\{5,5\}\) hyperbolic tiling called Bring's curve. Its qubits and stabilizer generators lie on the vertices of the small stellated dodecahedron. Admits a set of weight-five stabilizer generators.

Transversal Gates

Clifford group of four of the eight logical qubits can be done by transversal gates combined with qubit permutations [2].

Decoding

Fault-tolerant parity-check schedules whose performance is similar to those of the surface-17 code, but with qubit overhead reduced by a factor 2.6 [3].

Cousins

  • Polyhedron code— Bring code and related codes listed in [1; Table 1] arrange qubits and stabilizer generators on star polyhedra.
  • \([23, 12, 7]\) Golay code— The automorphism group of the parity-check matrix of the Golay code is the same as a certain automorphism group of the Bring code [2].
  • Dodecahedron code— The qubits and stabilizer generators of the \([[30,8,3]]\) Bring code lie on the vertices of the small stellated dodecahedron.
  • \([[9,1,3]]\) Surface-17 code— Bring's code and the surface-17 code have been compared numerically [1].

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
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum
Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Homological code
Hyperbolic surface code
2D hyperbolic surface code
Parents
2D hyperbolic surface code
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[30,8,3]]\) Bring code

References

[1]
J. Conrad, C. Chamberland, N. P. Breuckmann, and B. M. Terhal, “The small stellated dodecahedron code and friends”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, 20170323 (2018) arXiv:1712.07666 DOI
[2]
N. P. Breuckmann and S. Burton, “Fold-Transversal Clifford Gates for Quantum Codes”, Quantum 8, 1372 (2024) arXiv:2202.06647 DOI
[3]
J. Q. Broshuis, "The Small Stellated Dodecahedron Code: Finding Interleaved Measurement Schedules", Master's thesis. report-link GitHub-link
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Zoo Code ID: stellated_dodecahedron_css

Cite as:
“\([[30,8,3]]\) Bring code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/stellated_dodecahedron_css
BibTeX:
@incollection{eczoo_stellated_dodecahedron_css, title={\([[30,8,3]]\) Bring code}, booktitle={The Error Correction Zoo}, year={2025}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/stellated_dodecahedron_css} }
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Cite as:

“\([[30,8,3]]\) Bring code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/stellated_dodecahedron_css

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/surface/hyperbolic/2d_hyperbolic/stellated_dodecahedron_css.yml.