## Description

Two-dimensional lattice that exhibits optimal packing, solving the packing, kissing, covering and quantization problems. Its dual is the honeycomb lattice. The ruby lattice is a fattened honeycomb lattice interpolating between the honeycomb and hexagonal lattices.

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

## Parent

## Cousins

- Polygon code — The Voronoi cell of the hexagonal lattice is the hexagon.
- Universally optimal sphere packing — The hexagonal lattice code is universally optimal among all lattices, but has not been proven to be optimal over all periodic packings [11].
- Hexagonal GKP code
- XYZ ruby Floquet code — The XYZ ruby Floquet code is defined on the ruby lattice.
- Honeycomb Floquet code — The honeycomb Floquet code is defined on the honeycomb lattice.
- Truncated trihexagonal (4.6.12) color code — The 4.6.12 (truncated trihexagonal or square-hexagon-dodecagon) tiling is obtained by applying a fattening procedure to the triangular lattice [12].
- Honeycomb (6.6.6) color code — The triangular color code is defined on a trivalent lattice such as the honeycomb lattice.
- Kitaev honeycomb code — The Kitaev honeycomb model is defined on the honeycomb lattice.

## 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

## Page edit log

- Victor V. Albert (2022-11-23) — most recent

## Cite as:

“\(A_2\) hexagonal lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hexagonal