Efficient enumeration of quadratic lattices

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.

💡 Research Summary

The paper addresses the longstanding computational problem of enumerating all isometry classes of integral quadratic lattices of a prescribed rank (n) and determinant (D). While the classification of binary forms goes back to Gauss and tables for ranks up to five have been produced by Nipp and others, a general algorithm that works efficiently for arbitrary rank and determinant has been missing. The authors present a new algorithm that, given positive integers (n) and (D) with (\log D \ll n), outputs a lattice in every genus of rank (n) and determinant (D). Their main results are:

• Theorem 1.1 – an algorithm with running time
  \