\([[m 2^m / (m+1), 2^m / (m+1), 2\lfloor m/4 \rfloor + 1]]\) Khesin-Lu-Shor code

\([[m 2^m / (m+1), 2^m / (m+1), 2\lfloor m/4 \rfloor + 1]]\) 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].

Protection

The code distance is bounded above and conjectured to be \(m\) [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

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

Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum

Parents

Qubit CSS code

\([[m 2^m / (m+1), 2^m / (m+1), 2\lfloor m/4 \rfloor + 1]]\) Khesin-Lu-Shor code

Children

\([[6,2,2]]\) \(C_6\) code

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

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