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Hypercube code

Description

Spherical \((n,2^n,4/n)\) code whose codewords are vertices of an \(n\)-cube, i.e., all permutations and negations of the vector \((1,1,\cdots,1)\), up to normalization.

Cousins

Primary Hierarchy

Parents
Hypercube codewords in 2 (3, 4, \(n\)) dimensions form the vertices of a square (cube, tesseract, \(n\)-cube).
Hypercube codewords form the minimal lattice shell code of the \(\mathbb{Z}^n\) hypercubic lattice when the lattice is shifted such that the center of a hypercube is at the origin.
Hypercube codes form spherical 3-designs. The weighted union of the vertices of a hypercube and a cross polytope form a weighted spherical 5-design in dimensions \(\geq 3\) [1; Exam. 2.6].
Hypercube code

References

[1]
S. Borodachov, P. Boyvalenkov, P. Dragnev, D. Hardin, E. Saff, and M. Stoyanova, “Energy bounds for weighted spherical codes and designs via linear programming”, (2024) arXiv:2403.07457
[2]
A. B. Khesin, J. Z. Lu, and P. W. Shor, “Universal graph representation of stabilizer codes”, (2024) arXiv:2411.14448
[3]
N. Delfosse and B. W. Reichardt, “Short Shor-style syndrome sequences”, (2020) arXiv:2008.05051
[4]
P. Prabhu and B. W. Reichardt, “Distance-four quantum codes with combined postselection and error correction”, Physical Review A 110, (2024) arXiv:2112.03785 DOI
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Zoo Code ID: hypercube

Cite as:
“Hypercube code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/hypercube
BibTeX:
@incollection{eczoo_hypercube, title={Hypercube code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hypercube} }
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Permanent link:
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Cite as:

“Hypercube code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/hypercube

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/polytope/infinite/hypercube.yml.