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\(D_4\) hyper-diamond lattice

Description

BW lattice in dimension \(4\), which is the lattice corresponding to the \([4,1,4]\) repetition and \([4,3,2]\) SPC codes via Construction A. The lattice points form the \(\{3,3,4,3\}\) tesselation of 4-space [1; pg. 136].

Protection

The \(D_4\) lattice has a density of \(\pi^2/16\approx 0.6169\) and nominal coding gain of \(\sqrt{2}\). It exhibits the densest lattice packing in four dimensions [2].

Cousins

Member of code lists

Primary Hierarchy

Classical Domain
Analog Kingdom
Analog codeECC
Sphere packing
Lattice-based codeECC
Root lattice
\(D_n\) checkerboard latticeECC
Parents
\(D_n\) checkerboard lattice
Barnes-Wall (BW) latticeLattice-based ECC
\(D_4\) hyper-diamond lattice

References

[1]
H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.
[2]
A. Korkine and G. Zolotareff, “Sur les formes quadratiques positives quaternaires”, Mathematische Annalen 5, 581 (1872) DOI
[3]
T. Jochym-O’Connor and T. J. Yoder, “Four-dimensional toric code with non-Clifford transversal gates”, Physical Review Research 3, (2021) arXiv:2010.02238 DOI
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Zoo Code ID: dfour

Cite as:
“\(D_4\) hyper-diamond lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/dfour
BibTeX:
@incollection{eczoo_dfour, title={\(D_4\) hyper-diamond lattice}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/dfour} }
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Permanent link:
https://errorcorrectionzoo.org/c/dfour

Cite as:

“\(D_4\) hyper-diamond lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/dfour

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/analog/sphere_packing/lattice/root/dfour.yml.