What happens if we follow the construction of the Leech lattice but replace the relation
$\displaystyle \sum_{n=1}^{24} n^2 = 70^2$
with the smallest perfect squared square? Explicitly, if we set up a dot product on $\mathbf{R}^{22}$
$a \cdot b = a_1 b_1 + \ldots + a_{21} b_{21} - a_{22} b_{22}$
and consider the lattice II${}_{21,1}$ whose coordinates are all integers or half integers and
$a_1 + \ldots + a_{21} - a_{22}$
is even, then the lattice contains the vector
$v = (2, 4, 6, 7, 8, 9, 11, 15, 16, 17, 18, 19, 24, 25, 27, 29, 33, 35, 37, 42, 50, 112).$
(This vector comes from the smallest perfect squared square.)
Let $v^⊥ = ${ $a \in \mbox{II}_{21,1}$ | $a\cdot v = 0$ }; then $v$ is in $v^⊥$. The lattice $v^⊥/v$ is like a strange cousin of the Leech lattice. What's known about it?
