Midwest cousins of Barnes-Wall lattices

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 ²d−1±²d−k−1, for odd integers d?3 and integers . Their minimum norms are moderately high: .