# On the number of perfect lattices

## TL;DR

This paper establishes exponential bounds on the number of non-similar perfect lattices in high dimensions, showing that their count grows faster than any polynomial but slower than a double exponential.

## Contribution

It provides asymptotic exponential bounds on the number of perfect lattices in high dimensions, a significant advance in understanding lattice classification complexity.

## Key findings

- Lower bound: $p_d > e^{d^{1- ext{epsilon}}}$
- Upper bound: $p_d < e^{d^{3+ ext{epsilon}}}$
- Growth rate of perfect lattices is between polynomial and double exponential

## Abstract

We show that the number $p\_d$ of non-similar perfect $d$-dimensional lattices satisfies eventually the inequalities$e^{d^{1-\epsilon}}<p\_d<e^{d^{3+\epsilon}}$ for arbitrary smallstrictly positive $\epsilon$.