\(A_2\) hexagonal lattice code 


Two-dimensional lattice that exhibits optimal packing, solving the packing, kissing, covering and quantization problems. Its dual is the honeycomb 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.


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 \(G_2 = \frac{5}{36\sqrt{3}}\) [58]. It also solves the Gaussian channel coding problem [6].


Wireless communication [9,10].




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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
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
R. Kershner, “The Number of Circles Covering a Set”, American Journal of Mathematics 61, 665 (1939) DOI
L. Fejes Toth, Sur Ia representation d' une population infinie par une nombre fini d'elements, AMAH 10 (1959), 299-304
L. F. Tóth, Lagerungen in Der Ebene Auf Der Kugel Und Im Raum (Springer Berlin Heidelberg, 1972) DOI
A. Gersho, “Asymptotically optimal block quantization”, IEEE Transactions on Information Theory 25, 373 (1979) DOI
D. Newman, “The hexagon theorem”, IEEE Transactions on Information Theory 28, 137 (1982) DOI
C. Campopiano and B. Glazer, “A Coherent Digital Amplitude and Phase Modulation Scheme”, IEEE Transactions on Communications 10, 90 (1962) DOI
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
H. L. Montgomery, “Minimal theta functions”, Glasgow Mathematical Journal 30, 75 (1988) DOI
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 code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hexagonal
@incollection{eczoo_hexagonal, title={\(A_2\) hexagonal lattice code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hexagonal} }
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Cite as:

\(A_2\) hexagonal lattice code”, 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.