Efficient enumeration of quadratic lattices

TL;DR

This paper introduces an efficient algorithm for enumerating isometry classes of integral quadratic lattices, providing bounds on genus symbols and comparing practical performance with existing methods.

Contribution

It presents a new algorithm for lattice enumeration, analyzes its complexity, and offers publicly available implementations for improved efficiency.

Findings

The algorithm efficiently enumerates quadratic lattices.

Bounds on genus symbols are established.

Practical performance surpasses previous methods.

Abstract

We present an algorithm to enumerate isometry classes of integral quadratic lattices of a given rank and determinant, and analyze its running time by giving bounds on the number of genus symbols for a fixed rank and determinant. We build on previous work of Kirschmer, Brandhorst, Hanke, and Dubey and Holenstein. We analyze the running times of their respective algorithms and compare the practical performance of their implementations with our own. Our implementations are publicly available.