Generalized Theta Series of a Lattice

Mimicking the idea of the generalized Hamming weight of linear codes, we introduce a new lattice invariant, the generalized theta series. Applications range from identifying stable lattices to the lattice isomorphism problem. Moreover, we provide counterexamples for the secrecy gain conjecture on isodual lattices, which claims that the ratio of the theta series of an isodual (and more generally, formally unimodular) lattice by the theta series of the integer lattice $\mathbb{Z}^n$ is minimized at a (unique) symmetry point.