A Note on Optimal Unimodular Lattices
J.H. Conway, N.J.A. Sloane
TL;DR
This paper determines the maximum minimal norm of unimodular lattices up to dimension 33 and characterizes the existence of rootless unimodular lattices in certain dimensions, providing exact counts and classifications.
Contribution
It precisely identifies the highest minimal norm for unimodular lattices in dimensions up to 33 and classifies the existence of rootless lattices based on dimension.
Findings
Five odd 32-dimensional lattices achieve the highest minimal norm.
No rootless unimodular lattices exist in dimensions 23 and above, except in dimension 25.
Maximum minimal norm in dimension 33 exceeds 8×10^20.
Abstract
The highest possible minimal norm of a unimodular lattice is determined in dimensions n <= 33. There are precisely five odd 32-dimensional lattices with the highest possible minimal norm (compared with more than 8*10^20 in dimension 33). Unimodular lattices with no roots exist if and only if n >= 23, n not = 25.
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Taxonomy
TopicsAdvanced Algebra and Logic · Rings, Modules, and Algebras · Computability, Logic, AI Algorithms