Algebraic Constructions for Lattices

Abstract

In this chapter we construct dense lattice packings in dimensions 32, 36, 40, 48, 64, 96, ..., 65536, ... using a variety of techniques. The main constructions used are Constructions A_f (a more abstract version of Construction A), D (which uses a nested family of codes) and E (a powerful general construction which can be applied recursively). Other methods construct dense lattices from ideals in algebraic number fields. We also construct E ₈ and the Leech lattice using icosians.