Generalized Theta Series of a Lattice

Abstract

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.

arXiv commentPaper accepted for presentation at the 2025 IEEE Information Theory Workshop (ITW 2025). Final conference version