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\([[72,12,6]]\) half-gross code[1]

Alternative Names: \([[72,12,6]]\) BB6 code, \((6,6)\) BB6 code.

Description

A bivariate bicycle (BB) code with parameters \([[72,12,6]]\) and weight-six stabilizer generators [1].

One defining presentation uses \((\ell,m)=(6,6)\) with \(x^{\ell}=y^{m}=1\), and \(A=x^3+y+y^2\), \(B=y^3+x+x^2\) in \(\mathbb{F}_2[x,y]/(x^{\ell}-1,y^{m}-1)\) [1; Table 3].

Rate

Ancilla-added encoding rate is \(1/12\), using \(n_a=n=72\) ancilla qubits.

Decoding

Frontier decoder, a pruned dynamic-programming decoder, under circuit-level noise [2].

Cousin

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
Generalized homological-product qubit CSS codeQLDPC Qubit CSS Generalized homological-product Stabilizer Hamiltonian-based QECC Quantum
Bivariate bicycle (BB) codeQLDPC Qubit CSS Generalized homological-product Stabilizer Hamiltonian-based QECC Quantum
Parents
Bivariate bicycle (BB) code
2D lattice stabilizer codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum
The half-gross code belongs to the \((3,3)\)-BB family of 2D lattice stabilizer codes obtained by holding the check pattern fixed while enlarging the torus [1,4].
\([[72,12,6]]\) half-gross code

References

[1]
S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, “High-threshold and low-overhead fault-tolerant quantum memory”, Nature 627, 778 (2024) arXiv:2308.07915 DOI
[2]
A. Leverrier and R. Urbanke, “Approximating optimal decoding of quantum LDPC codes with narrow frontiers”, (2026) arXiv:2606.20513
[3]
V. V. Albert, unpublished computation, 2026
[4]
Z. Liang, B. Yang, J. T. Iosue, and Y.-A. Chen, “Operator algebra and algorithmic construction of boundaries and defects in (2+1)D topological Pauli stabilizer codes”, (2026) arXiv:2410.11942
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Zoo Code ID: bb72

Cite as:
“\([[72,12,6]]\) half-gross code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/bb72, arXiv:2606.11484
BibTeX:
@incollection{eczoo_bb72,
title={\([[72,12,6]]\) half-gross 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/bb72}
}
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Permanent link:
https://errorcorrectionzoo.org/c/bb72

Cite as:

“\([[72,12,6]]\) half-gross code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/bb72, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/bb72.yml.