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\([[9m-k,k,2]]_3\) triorthogonal code[1]

Description

Member of the \([[9m-k,k,2]]_3\) family of triorthogonal qutrit codes (for \(k\leq 3m-2\)) that admit a transversal diagonal gate in the third level of the qudit Clifford hierarchy and that are relevant for magic-state distillation.

Magic

For \(k = 3m-2\), the family yields the magic-state yield parameter \(\gamma = \log_2 (2+\frac{6}{3m-2}) \to 1\) as \(m\to\infty\) [1].

Member of code lists

Primary Hierarchy

Quantum Domain
Modular-qudit Kingdom
Modular-qudit codeQECC Quantum
Modular-qudit USt code
Modular-qudit stabilizer codeStabilizer Hamiltonian-based QECC Quantum
Modular-qudit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum
Prime-qudit triorthogonal codeCSS Stabilizer Hamiltonian-based QECC Quantum
Parents
Prime-qudit triorthogonal code
Small-distance block quantum codeQECC Quantum
\([[9m-k,k,2]]_3\) triorthogonal code

References

[1]
S. Prakash and T. Saha, “Low Overhead Qutrit Magic State Distillation”, Quantum 9, 1768 (2025) arXiv:2403.06228 DOI
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Zoo Code ID: qutrit_small_triorthogonal

Cite as:
“\([[9m-k,k,2]]_3\) triorthogonal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/qutrit_small_triorthogonal, arXiv:2606.11484
BibTeX:
@incollection{eczoo_qutrit_small_triorthogonal,
title={\([[9m-k,k,2]]_3\) 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/qutrit_small_triorthogonal}
}
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Permanent link:
https://errorcorrectionzoo.org/c/qutrit_small_triorthogonal

Cite as:

“\([[9m-k,k,2]]_3\) triorthogonal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/qutrit_small_triorthogonal, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/stabilizer/magic/qutrit_small_triorthogonal.yml.