Generalized Theta Series of a Lattice

TL;DR

This paper introduces a new lattice invariant called the generalized theta series, which helps analyze lattice properties and addresses conjectures related to isodual lattices and their secrecy gains.

Contribution

It defines the generalized theta series for lattices, explores its applications, and provides counterexamples to a conjecture on secrecy gains of isodual lattices.

Findings

Counterexamples to the secrecy gain conjecture for isodual lattices

Introduction of the generalized theta series as a new lattice invariant

Applications in identifying stable lattices and the lattice isomorphism problem

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.