  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 $2^{d-1}\pm 2^{d-k-1}$, for odd integers
$d\ge 3$ and integers $k=1,2, ..., \frac {d-1}2$. Their minimum norms are
moderately high: $2^{\lfloor \frac d2 \rfloor -1}$.

Number Theory;Representation Theory