The Leech lattice and other lattices

TL;DR

This paper explores the classification of unimodular lattices in various dimensions and introduces new constructions of unimodular lattices without roots in higher dimensions, advancing understanding of lattice structures.

Contribution

It provides a classification of unimodular lattices up to dimension 25 and constructs new unimodular lattices with no roots in dimensions 26 and 27.

Findings

Classified unimodular lattices in dimensions up to 25

Constructed unimodular lattices with no roots in dimensions 26 and 27

Enhanced understanding of lattice structures and their properties

Abstract

This is an unpublished manuscript written in 1983-4. It contains several results about lattices (=integral quadratic forms) including the classification of the unimodular lattices in dimensions up to 25 and the construction of unimodular lattices with no roots in dimensions 26 and 27.