Exact semidefinite programming bounds for packing problems

TL;DR

This paper develops an algorithm to convert semidefinite programming solutions into exact rational or quadratic solutions, enabling precise bounds for packing problems and proving uniqueness of optimal configurations like the E8 lattice.

Contribution

The authors introduce a method to round SDP solutions to exact algebraic solutions, leading to sharp bounds and uniqueness proofs for certain packing configurations.

Findings

Proved the E8 lattice configuration is uniquely optimal for minimal angular distance.

Established the sharpness of the three-point bound for specific spherical codes.

Computed exact upper bounds for sphere packing inside larger spheres.

Abstract

In this paper we give an algorithm to round the floating point output of a semidefinite programming solver to a solution over the rationals or a quadratic extension of the rationals. We apply this to get sharp bounds for packing problems, and we use these sharp bounds to prove that certain optimal packing configurations are unique up to rotations. In particular, we show that the configuration coming from the $\mathsf{E}_8$ root lattice is the unique optimal code with minimal angular distance $\pi/3$ on the hemisphere in $\mathbb R^8$, and we prove that the three-point bound for the $(3, 8, \vartheta)$-spherical code, where $\vartheta$ is such that $\cos \vartheta = (2\sqrt{2}-1)/7$, is sharp by rounding to $\mathbb Q[\sqrt{2}]$. We also use our machinery to compute sharp upper bounds on the number of spheres that can be packed into a larger sphere.