# Lattices with no roots and spread out shells

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.

• What's wrong with taking the E8 lattice and scaling it by a factor of $\sqrt{2}$ so that its shortest vectors have norm 4, followed by norm 8, and then norm 12, and so forth? – Adam P. Goucher Jul 12 at 12:49
• @AdamP.Goucher The E8 lattice has roots, the E8 root system. – M. Winter Jul 12 at 12:52