\([[m 2^m / (m+1), 2^m / (m+1), 2\lfloor m/4 \rfloor + 1]]\) Khesin-Lu-Shor code[1]
Description
A family of \([[m 2^m / (m+1), 2^m / (m+1),]]\) qubit CSS codes derived from the Hamming code. Their encoder-respecting form is the graph of a hypercube in \(m = 2^r - 1\) dimensions, and input nodes in the graph are codewords of the \([2^r-1,2^r-r-1,3]\) Hamming code [1].Cousins
- \([2^r-1,2^r-r-1,3]\) Hamming code— The encoder-respecting form of the \([[m 2^m / (m+1), 2^m / (m+1), 2\lfloor m/4 \rfloor + 1]]\) Khesin-Lu-Shor code is the graph of a hypercube in \(m = 2^r - 1\) dimensions, and input nodes in the graph are codewords of the \([2^r-1,2^r-r-1,3]\) Hamming code [1].
- Hypercube code— The encoder-respecting form of the \([[m 2^m / (m+1), 2^m / (m+1), 2\lfloor m/4 \rfloor + 1]]\) Khesin-Lu-Shor code is the graph of a hypercube in \(m = 2^r - 1\) dimensions, and input nodes in the graph are codewords of the \([2^r-1,2^r-r-1,3]\) Hamming code [1].
- \([[7,1,3]]\) Steane code— The encoder-respecting form of both the Steane and Khesin-Lu-Shor codes is the graph of a hypercube [1].
Member of code lists
Primary Hierarchy
Parents
\([[m 2^m / (m+1), 2^m / (m+1), 2\lfloor m/4 \rfloor + 1]]\) Khesin-Lu-Shor code
Children
The Khesin-Lu-Shor code for \(r=2\) and \(m=2^r - 1 = 3\) is the \(C_6\) code.
References
- [1]
- A. B. Khesin, J. Z. Lu, and P. W. Shor, “Universal graph representation of stabilizer codes”, (2025) arXiv:2411.14448
Page edit log
- Victor V. Albert (2024-12-11) — most recent
- Andrey Boris Khesin (2024-12-11)
Cite as:
“\([[m 2^m / (m+1), 2^m / (m+1), 2\lfloor m/4 \rfloor + 1]]\) Khesin-Lu-Shor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/kls