I am looking for lattices with the following properties:

- The lattice has no roots.
- The norm (squared length) of the second shortest vectors should be at least twice as large as the norm of the shortest vector. If the factor is smaller than $2$ but close to it, I am probably still interested.

I do not care whether the lattice is unimodular or integral. However, the following properties are favorable:

- The lattice gives a spherical code with small spherical covering radius. That is, no point on the first shell is too far away from the lattice. For example, in the Leech lattice of minimum length $2$ you got (if I am not mistaken) a spherical covering radius of $\sqrt{5/2}$, and I consider this as small.
- The lattice has a large automorphism group. For example, it is transitive on shortest vectors, or transitive on pairs of shortest vectors that are a certain distance apart while fixing the origin.

Basically, the Leech lattice would be a perfect candidate, but there is a shell of norm $6$ between the norm-$4$ and norm-$8$ vectors.