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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?

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A third edition of Lattices and Codes by Wolfgang Ebeling has appeared, or will soon: springer.com/springer+spektrum/mathematik/book/… From section 4.4 in the second edition, it is not necessary for the numbers to be squared be distinct, as a string of 1's followed by a 3 gives $\mathbb E_8.$ Your lattice is not unimodular so the usual suspects have not written about it explicitly. But take a look at math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/index.html#20D – Will Jagy Nov 27 '12 at 21:03

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