Sphere packings of dimension 8

Introduction

A page is devoted to eight-dimensional sphere packings since these are geometrically particularly appealing and have found usage in optical communication systems. The dimensions may in a communication scenario represent quadratures, time slots, frequency bands, or polarizations, as discussed on the page for other dimensions.

The best lattice in eight dimensions (8d) is the E₈ lattice. It can be constructed from the 8d cubic lattice by shifting every second point along the main diagonal, halfway to the next point in the cubic lattice. This shifting operation increases the minimum distance by √2 without changing the density. Thus, the coding gain of E₈ over the cubic lattice is a factor of 2, or 3.01 dB. The total gain, which also includes a shaping gain, is 10log₁₀((5π/6)(2/3)^1/4) = 3.739 dB [Calderbank87, Sec. IV], [Conway99, pp. 73–74, 121].

Subsets of the E₈ lattice have been designed by numerical optimization for sizes up to 256 [Agrell14] and selected larger sizes [Agrell16]. These are included under the name l8_*.txt. We conjecture that these packings are optimal lattice subsets for all included sizes, however with no claim that they would be optimal among nonlattice packings. Only one case has been found when a nonlattice packing is better than the best E₈-based packing, which occurs for M = 10.

The most power-efficient 8d packing known, in the sense of maximizing γ, is the biorthogonal packing with M = 16 and γ = 3.010 dB. As M increases, γ is irregular and displays two other prominent peaks with almost the same power efficiency, which are 3.007 dB at M = 58 and 2.981 dB at M = 241. As M continues to increase, the power efficiency drops rapidly [Agrell14]. At the same time, the gain G increases slowly towards 3.739 dB (see above), but it seems like very large sizes would be needed to get near this asymptotic value.

Database

References

[Agrell14]

E. Agrell, unpublished, 2014.

[Agrell16]

E. Agrell, unpublished, 2016.