Improved sphere packing counts found in high dimensions 25 to 31
New lower bounds for kissing numbers in dimensions $25$--$29$ and $31$
Information Theory
Summary
The kissing number problem asks how many same-sized spheres can touch one central sphere without overlapping. The paper improves known minimum counts for this problem in high dimensions from 25 to 31. The authors build on existing arrangements in dimension 24 and use geometric rotations and deformations to fit more spheres around the center. This work updates the best known lower limits on how many spheres can simultaneously touch the center sphere in these specific dimensions.
What this means in practice
- •For coding theory engineers: Use improved sphere packing bounds to optimize error-correcting code design for high-dimensional signal spaces.
- •For cryptography system designers: Incorporate better high-dimensional geometric packing limits to develop stronger lattice-based cryptographic schemes.
Authors
Rustem Takhanov, Stanislav Yun
Abstract
The kissing number in dimension $d$ is the largest number of non-overlapping congruent spheres that can simultaneously touch a central sphere of the same size. We study dimensions $25$-$31$, where the best previous constructions are based on Leech lifting from the optimal kissing configuration in dimension $24$. Our method exploits the absence of contacts between the unlifted bulk and the block consisting of lifted and auxiliary vectors. Rotating this block while keeping the bulk fixed creates room for two antipodal points in dimensions $26$, $27$, and $28$, and one point in dimension $29$. Two further modifications yield improvements in dimensions $25$ and $31$: a nonorthogonal diagonal linear deformation of the lifted block admits two antipodal points in dimension $25$, while rotating only the additional coordinates of the lifted vectors admits four nonantipodal points in dimension $31$. Together, these constructions yield the new lower bounds $τ_{25}\geq 197058$, $τ_{26}\geq 198552$, $τ_{27}\geq 200046$, $τ_{28}\geq 204522$, $τ_{29}\geq 209497$, and $τ_{31}\geq 238354$.