Genus distribution of random $q$-ary lattices

The genus is an efficiently computable arithmetic invariant for lattices up to isomorphism. Given the recent proposals of basing cryptography on the lattice isomorphism problem, it is of cryptographic interest to classify relevant families of lattices according to their genus. We propose such a classification for $q$-ary lattices, and also study their distribution. In particular, for an odd prime $q$, we show that random $q$-ary lattices are mostly concentrated on two genera. Because the genus is local, this also provides information on the distribution for general odd $q$. The case of $q$ a power of 2 is also studied, although we only achieve a partial classification.