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\([[49,1,5]]\) triorthogonal code[1; Appx. B]

Description

Triorthogonal and quantum divisible code which is the smallest distance-five stabilizer code to admit a transversal \(T\) gate [2][1; Appx. B]. It is one example of a level-three generalized divisible code obtainable from the doubling transformation [3; Sec. VI.D]. Its \(X\)-type stabilizers form a triply even linear binary code in the symplectic representation.

Magic

The code yields an exponent \(\gamma = \log 49 / \log 5 \approx 2.42\).

Transversal and Permutation-Based Gates

The code admits a transversal \(T\) gate [1; Appx. B].

Cousins

  • Doubled color code— The \([[49,1,5]]\) triorthogonal code can be viewed as a (gauge-fixed) doubled color code obtained from the \([[17,1,5]]\) 4.8.8 color code via the doubling transformation [4].
  • \([[17,1,5]]\) 4.8.8 color code— The \([[49,1,5]]\) triorthogonal code can be viewed as a (gauge-fixed) doubled color code obtained from the \([[17,1,5]]\) 4.8.8 color code via the doubling transformation [4].
  • Divisible code— The \([[49,1,5]]\) triorthogonal code stabilizer generator matrix can be obtained from a triply even linear binary code [1; Appx. B].
  • Binary dihedral PI code— The \(((27,2,5))\) binary dihedral PI code realizes the \(T\) gate (strongly) transversally, but requires fewer qubits than the \([[49,1,5]]\) triorthogonal code.

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
\(k\)-orthogonal code
Triorthogonal codeCSS Stabilizer Hamiltonian-based QECC Quantum
Parents
Triorthogonal code
Quantum divisible codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[49,1,5]]\) triorthogonal code

References

[1]
S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI
[2]
K. Betsumiya and A. Munemasa, “On triply even binary codes”, Journal of the London Mathematical Society 86, 1 (2012) arXiv:1012.4134 DOI
[3]
J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
[4]
S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
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Zoo Code ID: stab_49_1_5

Cite as:
“\([[49,1,5]]\) triorthogonal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_49_1_5, arXiv:2606.11484
BibTeX:
@incollection{eczoo_stab_49_1_5,
title={\([[49,1,5]]\) triorthogonal code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/stab_49_1_5}
}
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Permanent link:
https://errorcorrectionzoo.org/c/stab_49_1_5

Cite as:

“\([[49,1,5]]\) triorthogonal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_49_1_5, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/magic/k-orthogonal/stab_49_1_5.yml.