A study on some approximations on the average number of the LLL bases in higher dimensions

TL;DR

This paper investigates approximations for the average number of LLL bases in high dimensions, providing practical formulas that simplify computation compared to existing complex theoretical formulas.

Contribution

It introduces new approximation methods for calculating the average number of LLL bases in high dimensions, making the process more computationally feasible.

Findings

Proposed simple exponential-based approximation formulas

Validated approximations reduce computational complexity

Applicable to high-dimensional lattice basis analysis

Abstract

There is a result related to the average number of the $(\delta, \eta)$-LLL bases in dimension $n$ in theoretical sense but the formula seems to be complicated and computing in high dimension takes a long time. In practical sense, we suggest some approximations which can be computed by just storing some constants and computing relatively simple exponential functions.