# Midwest cousins of Barnes-Wall lattices - Mathematics > Number Theory

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Abstract: Given a rational lattice and suitable set of linear transformations, weconstruct a cousin lattice. Sufficient conditions are given for integrality,evenness and unimodularity. When the input is a Barnes-Wall lattice, we getmulti-parameter series of cousins. There is a subseries consisting ofunimodular 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 aremoderately high: $2^{\lfloor \frac d2 floor -1}$.

Author: Robert L. Griess Jr

Source: https://arxiv.org/