[Jump to code hierarchy]

\([[22,8,4]]\) Grassl-Rötteler code[1,2]

Description

Self-dual pure CSS code constructed from a self-orthogonal \([22,7,8]\) classical binary code [1; Sec. IV.C]. All of its logical Clifford gates can be realized by depth-one circuits of two-local code-preserving physical gates [2].

The underlying classical code has weight enumerator \(1 + 21y^{8} + 42y^{10} + 42y^{12} + 21y^{14} + y^{22}\); it is even but not doubly even, and it contains the all-ones vector. Its permutation automorphism group is isomorphic to \(PGL(2,7)\), of order 336 [1; Sec. IV.C], fixing one distinguished qubit and acting transitively on the remaining 21.

Both an \(X\)-type and a \(Z\)-type stabilizer generator are supported on each row of the following stabilizer tableau [2][3; ID 67374955079675676c5c196a] \begin{align} \begin{smallmatrix} Z & I & I & I & I & I & I & I & I & Z & Z & Z & Z & Z & Z & I & I & Z & Z & Z & Z & Z \\ I & Z & I & I & I & I & I & Z & Z & I & I & I & I & Z & Z & I & Z & I & Z & Z & Z & Z \\ I & I & Z & I & I & I & I & Z & I & Z & I & Z & Z & I & Z & Z & Z & Z & I & I & I & Z \\ I & I & I & Z & I & I & I & I & Z & I & Z & I & I & I & Z & Z & Z & Z & Z & Z & Z & I \\ I & I & I & I & Z & I & I & Z & I & I & I & Z & I & Z & Z & Z & Z & Z & Z & Z & I & I \\ I & I & I & I & I & Z & I & Z & I & I & Z & Z & Z & I & I & Z & Z & Z & Z & I & Z & I \\ I & I & I & I & I & I & Z & Z & Z & Z & I & Z & I & I & I & Z & I & I & I & Z & Z & I \end{smallmatrix}~. \tag*{(1)}\end{align}

Transversal and Permutation-Based Gates

Qubit permutations combined with transversal CNOT gates between two code blocks realize the full logical linear group \(GL(16,2)\) on the 16 logical qubits of the two blocks [1; Sec. IV.C].All logical Clifford gates can be realized as two-fold transversal gates, i.e., by depth-one circuits of two-local code-preserving physical gates [2].

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 Hamiltonian-based QECC Quantum
Self-dual CSS codeStabilizer Hamiltonian-based Qubit QECC Quantum
Parents
Self-dual CSS code
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[22,8,4]]\) Grassl-Rötteler code

References

[1]
M. Grassl and M. Roetteler, “Leveraging automorphisms of quantum codes for fault-tolerant quantum computation”, 2013 IEEE International Symposium on Information Theory 534 (2013) arXiv:1302.1035 DOI
[2]
V. V. Albert, “Beyond transversality: structure of Clifford circuits for CSS codes”, (2026) arXiv:2608.05688
[3]
S. Burton, “qecdb.org: Quantum Error Correction Database”, URL
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)

— see instructions

Zoo Code ID: stab_22_8_4

Cite as:
“\([[22,8,4]]\) Grassl-Rötteler code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_22_8_4, arXiv:2606.11484
BibTeX:
@incollection{eczoo_stab_22_8_4,
title={\([[22,8,4]]\) Grassl-Rötteler 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_22_8_4}
}
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/stab_22_8_4

Cite as:

“\([[22,8,4]]\) Grassl-Rötteler code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_22_8_4, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/22/stab_22_8_4.yml.