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\([[2^r-1, 2^r-2r-1, 3]]_p\) quantum Hamming code[1]

Description

A family of CSS codes extending quantum Hamming codes to prime qudits of dimension \(p\) by expressing the qubit code stabilizers in local-dimension-invariant (LDI) form [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 RM codeStabilizer Hamiltonian-based QECC Quantum
Parents
Prime-qudit RM code
\([[2^r-1, 2^r-2r-1, 3]]_p\) quantum Hamming codes are constructed using \(q\)-ary Hamming codes, which themselves are dual to first-order GRM codes [2; pg. 45].
Small-distance block quantum codeQECC Quantum
\([[2^r-1, 2^r-2r-1, 3]]_p\) quantum Hamming code
Children
\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code
\([[2^r-1, 2^r-2r-1, 3]]_p\) prime-qudit CSS code for \(p=2\) reduce to \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming codes.

References

[1]
A. J. Moorthy and L. G. Gunderman, “Local-dimension-invariant Calderbank-Shor-Steane Codes with an Improved Distance Promise”, (2021) arXiv:2110.11510
[2]
M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes (Springer Netherlands, 1991) DOI
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Zoo Code ID: qudit_hamming_css

Cite as:
“\([[2^r-1, 2^r-2r-1, 3]]_p\) quantum Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_hamming_css
BibTeX:
@incollection{eczoo_qudit_hamming_css, title={\([[2^r-1, 2^r-2r-1, 3]]_p\) quantum Hamming code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qudit_hamming_css} }
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Permanent link:
https://errorcorrectionzoo.org/c/qudit_hamming_css

Cite as:

“\([[2^r-1, 2^r-2r-1, 3]]_p\) quantum Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_hamming_css

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