I'm studying Venkov's proof of the classification of even unimodular rank 24 lattices, and it prompted the following question.


For an even unimodular lattice $L$, let $R(L)= \{ x \in L : (x,x) =2\}$ be its root system. 


**Question.** Can one "roughly" classify, up to isomorphism, all rank $32$ even unimodular lattices with root system $32 A\_1$? Or are there too many of these? (Note: there are at least $10^8$ even unimodular lattices of rank 32.) 


*Remark.* There are no even unimodular lattices of rank 8 and 16 with root system $8 A\_1$ and $16A\_1$, respectively. There is a unique rank $24$ even unimodular lattice with root system $24 A\_1$ by the work of Niemeier (and Venkov).


Any comments are appreciated!



Even unimodular lattices with root system $32 A_1$
Even unimodular lattices of rank $n$ with root system $n\cdot A\_1$
correspond bijectively with Type II codes of length $n$ and minimal
weight at least $8$. ("Type II" = self-dual and doubly even.)
The results you quote for $n=8,16,24$ correspond to the fact that
there's no such code for $n=8$ and $n=16$,
and a unique one for $n=24$ (the extended binary Golay code).
For $n=32$ there are five such codes according to



> 
> John H. Conway and Vera S. Pless, On the enumeration of self-dual codes,
>  *J. Combin. Theory Ser. A* **28** (1980), 26$-$53.
> 
> 
> 


Therefore there are $5$ even unimodulars of rank $32$ with roots $32A\_1$.
For $n=40$ the count is
[at least 17493](http://llama.mshri.on.ca/~king/ok-codes.pdf)
(Oliver King, 2001).


The correspondence is via what's called "Construction A" in SPLAG =



> 
> John H. Conway and Neil J. A. Sloane,
>  *Sphere Packings, Lattices and Groups*.
>  New York: Springer-Verlag, 1998.
> 
> 
> 


This construction associates to any length-$n$ binary linear code $C$
the lattice
$$
L\_C := 2^{-1/2} \{ v \in {\bf Z}^n : v \bmod 2 \in C \}.
$$
This lattice is even unimodular *iff* $C$ is Type II;
then the root system contains $n \cdot A\_1$,
and is exactly $n \cdot A\_1$ *iff* $C$ has no vectors of weight $4$.


To recover $C$ from $L$:
Given an integral lattice $L$ of rank $n$
with root system containing $n \cdot A\_1$,
we have $L \subseteq L^\* \subseteq (n \cdot A\_1)^\* = n \cdot \frac12 A\_1$.
Lattices sandwiched between $n \cdot \frac12 A\_1$ and $n \cdot A\_1$
correspond bijectively with subgroups of the quotient group
$(n \cdot \frac12 A\_1) / (n \cdot A\_1) \cong ({\bf Z}/2{\bf Z})^n$.
A subgroup of $({\bf Z}/2{\bf Z})^n$ is a linear code $C$,
and we soon confirm that $L \cong L\_C$.


