  Given a polarization of an even unimodular lattice and integer $k\ge 1$, we
define a family of unimodular lattices $L(M,N,k)$. Of special interest are
certain $L(M,N,3)$ of rank 72. Their minimum norms lie in $\{4, 6, 8\}$. Norms
4 and 6 do occur. Consequently, 6 becomes the highest known minimum norm for
rank 72 even unimodular lattices. We discuss how norm 8 might occur for such a
$L(M,N,3)$. We note a few $L(M,N,k)$ in dimensions 96, 120 and 128 with
moderately high minimum norms.
