- The paper presents significant improvements to kissing numbers in dimensions 17–21, achieving lower bounds of at least 5730, 7654, 11692, 19448, and 29768.
- It employs a novel, computer-free methodology that modifies lattice vector sign conventions to create additional spatial configurations.
- The research opens avenues for applications in coding theory and further investigation into optimal sphere packings in high-dimensional spaces.
Improved Kissing Numbers in Seventeen Through Twenty-One Dimensions
This paper, authored by Henry Cohn and Anqi Li, presents significant advancements in the field of discrete geometry, specifically concerning the determination of kissing numbers in higher dimensions. The kissing number is a classic problem that asks for the maximum number of non-overlapping unit spheres that can be arranged such that they all touch another unit sphere. This issue can be conceptualized as a sphere packing problem within spherical spaces or an equivalent problem within Euclidean spaces. Historically, the exact solutions to kissing numbers are known only for dimensions 1, 2, 3, 4, 8, and 24, with significant developments needed for other dimensions.
The authors of this paper focus on improving the known lower bounds for dimensions 17 through 21, which have stood unchanged since the work of Leech in 1967. The paper improves these lower bounds by significant amounts: 384, 256, 1024, 2048, and 2048 in dimensions 17, 18, 19, 20, and 21, respectively, resulting in kissing numbers of at least 5730, 7654, 11692, 19448, and 29768. Cohn and Li achieved these improvements through a novel approach that modifies the signs in the known lattice vector constructions to create additional space, an avenue not previously explored in reaching these bounds.
The methodology employed does not rely on computer-based calculations, thus showcasing a different analytical approach. However, to ensure validation and reproducibility, the authors have made available files containing explicit coordinates for these configurations, which can be checked algorithmically.
A particular highlight within this work is the treatment of dimensions 19 through 21, where a simpler construction is outlined. The authors utilize binary error-correcting codes and structural adjustments based on vector permutations to construct valid kissing configurations. This construction was extended from previously known configurations and succeeded in adding more spheres by employing vectors orthogonal to the existing setup. Furthermore, when exploring dimensions 17 and 18, the authors employed more intricate techniques extending from a known 16-dimensional configuration, showing once more the ability to innovate within existing frameworks.
Theoretical implications of this research indicate that the laminated lattices, which contain the prior bounds, may be suboptimal for these dimensions. As a result, the paper suggests the potential room for even further improvements in other dimensions, especially considering that the approach in 24-dimensional space relies heavily on a different type of lattice, the Leech lattice.
Practically, this work not only enhances the understanding of geometric configurations in high-dimensional spaces but also hints at applications in coding theory and information sciences where such geometric problems find utility in signal processing and data transmission. The advancement also shows promise for extending techniques into dimensions 22 and possibly 23, although no successful augmentation has yet been made for these higher dimensions using the current methods.
This research contributes to both the theoretical landscape of mathematics and to potential practical applications in computational simulations, coding theory, and probabilistic models. Future advances could emerge from refining current techniques or exploring new geometrical insights, perhaps leading to further records in the literature of sphere packings and configurations in high-dimensional spaces.