Lattices (resp. codes) achieving equality are called

For these , there are just two extremal codes known, the extended quadratic residue codes of length 24 and 48, both are the unique extremal codes in their length. One intensively studied question is the existence of an extremal code of length 72. Using theoretical and computational methods one may show that the automorphism group of such an extremal code is rather small: its order is either 5 or divides 24.

The Leech lattice is the unique extremal lattice of dimension 24, in dimension 48
one knows 3 extremal lattices and there is at least one of dimension 72.
It is an interesting question whether there are other extremal lattices of dimension 48.
I will report on methods to narrow down the possible automorphisms of such lattices
and on number theoretic computations to classify all lattices with an automorphism of
order with
.