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\([[2^r+r, 2^r-r-2, 3]]\) Ring CPC code[1]

Description

A family of \([[2^r+r, 2^r-r-2, 3]]\) CPC codes for \(r \geq 3\) whose matrices are based on the shortened version of the \([2^r-1,2^r-r-1,3]\) Hamming code. See [1; Thm. 4] for their stabilizer generator matrix.

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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
Parents
Coherent-parity-check (CPC) code
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[2^r+r, 2^r-r-2, 3]]\) Ring CPC code

References

[1]
N. Chancellor, A. Kissinger, S. Zohren, J. Roffe, and D. Horsman, “Graphical structures for design and verification of quantum error correction”, Quantum Science and Technology 8, 045028 (2023) arXiv:1611.08012 DOI
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Zoo Code ID: ring_cpc

Cite as:
“\([[2^r+r, 2^r-r-2, 3]]\) Ring CPC code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/ring_cpc, arXiv:2606.11484
BibTeX:
@incollection{eczoo_ring_cpc,
title={\([[2^r+r, 2^r-r-2, 3]]\) Ring CPC 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/ring_cpc}
}
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Permanent link:
https://errorcorrectionzoo.org/c/ring_cpc

Cite as:

“\([[2^r+r, 2^r-r-2, 3]]\) Ring CPC code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/ring_cpc, arXiv:2606.11484

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