\([[2^r-1, 2^r-2r-1, 3]]_p\) 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].
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 code
Child
- \([[2^r-1, 2^r-2r-1, 3]]\) 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
Page edit log
- Victor V. Albert (2022-02-10) — most recent
- Lane G. Gunderman (2022-02-10)
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