\([[22,2,6]]\) shortened Golay code

\([[22,2,6]]\) shortened Golay code[1]

Description

Self-dual doubly even CSS code on 22 physical qubits encoding two logical qubits with distance six. The code is constructed from the self-orthogonal \([22,10,8]\) code obtained by shortening the \([24,12,8]\) extended Golay code on two bits [1; Appx. A.1.5]. Since the automorphism group of the extended Golay code is 5-transitive, the code does not depend on which two bits are chosen [1]. The code is also a symplectic double of a pure \([[11,1,3]]\) stabilizer code.

Protection

Detects errors on up to five qubits and corrects errors on up to two qubits.

Transversal and Permutation-Based Gates

Code automorphisms and fold-transversal gates generate the full two-qubit Clifford group [2].

Cousins

\([[23, 1, 7]]\) Quantum Golay code— The \([[22,2,6]]\) shortened Golay code is obtained by shortening the extended Golay code on two bits, while the qubit Golay code is obtained by shortening on one bit [1; Appx. A.1.5].
\([[11,1,5]]_4\) Galois-qudit CSS code— Binarizing the \([[11,1,5]]_4\) code in the self-dual normal basis \(\{\omega,\omega^2\}\) yields the \([[22,2,6]]\) shortened Golay code, raising the distance from five to six.

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeQubit Stabilizer Hamiltonian-based QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum

Maximal Cube Root (MCR) codeCSS Generalized homological-product Lattice stabilizer Stabilizer Hamiltonian-based QECC Quantum

Parents

Maximal Cube Root (MCR) code

The \([[22,2,6]]\) shortened Golay code is permutation-equivalent to the MCR code with \(\ell=11\), \(f=x+1\), \(p=1\), and \(q=x^9+x^5+x^4+x^3+x\) [2].

Self-dual CSS codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum

The \([[22,2,6]]\) shortened Golay code is the self-dual CSS code of the doubly even self-orthogonal \([22,10,8]\) code obtained by shortening the extended Golay code on two bits [1; Appx. A.1.5].

Bipartite cyclic cluster (BCC) codeQubit CSS Stabilizer Hamiltonian-based QECC Quantum

The \([[22,2,6]]\) shortened Golay code is permutation-equivalent to a BCC code with \(\mathcal{S}=\{1,3,5,9,15\}\), saturating the BCC distance bound \(d\leq|\mathcal{S}|+1\) [3].

\([[22,2,6]]\) shortened Golay code

References

[1]: J. Haah, M. B. Hastings, D. Poulin, and D. Wecker, “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
[2]: A. Davenport, J. Blue, and I. Chuang, “Generalized Bicycle Codes as Cyclic Submodules and their Automorphism Structure”, (2026) arXiv:2606.05044
[3]: M. B. Hastings, “A Class of Cyclic Quantum Codes”, (2025) arXiv:2509.06865

Page edit log

Victor V. Albert (2026-08-28) — most recent
Shubham P. Jain (2026-08-28)

Cite as:

“\([[22,2,6]]\) shortened Golay code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_22_2_6, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/generalized_bicycle/stab_22_2_6.yml.