A New Bound for the Orthogonality Defect of HKZ Reduced Lattices

TL;DR

This paper establishes a new, sharper upper bound on the orthogonality defect of HKZ reduced lattices, improving understanding of lattice basis quality especially in low dimensions and extending to arbitrary ranks.

Contribution

It provides the first sharp bound for dimensions up to 3 and a general improved bound for any rank, surpassing previous results by Lagarias, Lenstra, and Schnorr.

Findings

Sharp upper bound for dimension 3

General upper bound for arbitrary rank

Improved bounds over previous literature

Abstract

In this work, we determine a sharp upper bound on the orthogonality defect of HKZ reduced bases up to dimension $3$. Using this result, we determine a general upper bound for the orthogonality defect of HKZ reduced bases of arbitrary rank. This upper bound seems to be sharper than existing bounds in literature, such as the one determined by Lagarias, Lenstra and Schnorr.