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: .