Midwest cousins of Barnes–Wall lattices

Abstract

Given a rational lattice and suitable set of linear transformations, we construct a cousin lattice. Sufficient conditions are given for integrality, evenness and unimodularity. When the input is a Barnes–Wall lattice, we get multi-parameter series of cousins. There is a subseries consisting of unimodular lattices which have ranks 2 d − 1 ± 2 d − k − 1 , for odd integers d ⩾ 3 and integers k = 1 , 2 , … , d − 1 2 . Their minimum norms are moderately high: 2 ⌊ d 2 ⌋ − 1 .