Minkowski bases, Korkin-Zolotarev bases and successive minima

TL;DR

This paper investigates properties of lattice bases, proving new bounds on basis vector lengths, confirming conjectures, and constructing examples that challenge existing assumptions about lattice basis optimality.

Contribution

It establishes new bounds on Minkowski-reduced basis vectors, confirms a conjecture of Schürmann, and constructs lattices with unusual basis properties.

Findings

Proved $ vert v_k vert^2 \\leq \frac{k}{4} \\lambda_k^2$ for $k=6,7$

Constructed lattices where basis vectors are longer than the longest vector in Korkin-Zolotarev bases

Found lattices where bases containing the shortest vector are not the shortest bases

Abstract

Let $\lambda_k$ denote the $k$-th successive minimum of a lattice $L$. We study properties of the lengths of certain bases of $L$. If $v_1, \dots v_n$ is a basis which is reduced in the sense of Minkowski we show that $\lvert v_k \rvert^2 \leq \frac{k}{4} \lambda_{k}^2$ for $k = 6, 7$, confirming a conjecture of Sch\"urmann, and obtaining the first improvement of a classical bound by Van der Waerden. We construct a sequences of lattices where $\lvert v_n \rvert$ is significantly longer than the longest vector in a Korkin-Zolotarev reduced basis, answering a question of Sch\"urmann. In an appendix joint with Lior Hadassi we construct a lattice $L$ with the surprising property that any basis containing the shortest vector of $L$ is not the shortest basis.