\(A_2\) hexagonal lattice 

Description

Two-dimensional lattice that exhibits optimal packing, solving the packing, kissing, covering and quantization problems. Its dual is the honeycomb tiling, which is not a lattice (since the points do not form a group under addition) but which consists of two hexagonal lattices. The ruby lattice is a fattened honeycomb tiling interpolating between the honeycomb tiling and hexagonal lattice.

It's generator matrix is \begin{align} V=\begin{pmatrix}1 & 0\\ -1/2 & \sqrt{3}/2 \end{pmatrix}~. \tag*{(1)}\end{align} All possible sublattices are characterized in Refs. [1,2] from the point of view of information transmission over the AGWN channel.

Protection

The hexagonal lattice exhibits the densest packing with density \(\Delta = \pi/\sqrt{12} \approx 0.9069\) [3; Sec. 1.4], the highest kissing number of 6, and the thinnest covering with thickness \(\Theta = 2\pi/3\sqrt{3}\approx 1.2092\) [4] in two dimensions. It solves the quantizer problem in two dimensions with \(G_2 = \frac{5}{36\sqrt{3}}\) [5–8]. It also solves the Gaussian channel coding problem [6].

Realizations

Wireless communication [9,10].

Parent

Cousins

References

[1]
M. Bernstein, N. J. A. Sloane, and P. E. Wright, “On sublattices of the hexagonal lattice”, Discrete Mathematics 170, 29 (1997) DOI
[2]
M. Baake and P. A. B. Pleasants, “Algebraic Solution of the Coincidence Problem in Two and Three Dimensions”, Zeitschrift für Naturforschung A 50, 711 (1995) DOI
[3]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[4]
R. Kershner, “The Number of Circles Covering a Set”, American Journal of Mathematics 61, 665 (1939) DOI
[5]
L. Fejes Toth, Sur Ia representation d' une population infinie par une nombre fini d'elements, AMAH 10 (1959), 299-304
[6]
L. F. Tóth, Lagerungen in Der Ebene Auf Der Kugel Und Im Raum (Springer Berlin Heidelberg, 1972) DOI
[7]
A. Gersho, “Asymptotically optimal block quantization”, IEEE Transactions on Information Theory 25, 373 (1979) DOI
[8]
D. Newman, “The hexagon theorem”, IEEE Transactions on Information Theory 28, 137 (1982) DOI
[9]
C. Campopiano and B. Glazer, “A Coherent Digital Amplitude and Phase Modulation Scheme”, IEEE Transactions on Communications 10, 90 (1962) DOI
[10]
G. Foschini, R. Gitlin, and S. Weinstein, “Optimization of Two-Dimensional Signal Constellations in the Presence of Gaussian Noise”, IEEE Transactions on Communications 22, 28 (1974) DOI
[11]
H. L. Montgomery, “Minimal theta functions”, Glasgow Mathematical Journal 30, 75 (1988) DOI
[12]
H. Bombin and M. A. Martin-Delgado, “Exact topological quantum order inD=3and beyond: Branyons and brane-net condensates”, Physical Review B 75, (2007) arXiv:cond-mat/0607736 DOI
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Zoo Code ID: hexagonal

Cite as:
“\(A_2\) hexagonal lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hexagonal
BibTeX:
@incollection{eczoo_hexagonal, title={\(A_2\) hexagonal lattice}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hexagonal} }
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“\(A_2\) hexagonal lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hexagonal

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