Formally Unimodular Packings for the Gaussian Wiretap Channel

TL;DR

This paper introduces formally unimodular packings, a generalization of lattices, and demonstrates their effectiveness in enhancing secrecy gain in Gaussian wiretap channels through a universal approach based on code weight distributions.

Contribution

It develops the theory of formally unimodular packings, links them to self-dual codes, and shows they can outperform existing unimodular lattices in secrecy gain for wiretap channels.

Findings

Formally unimodular packings share secrecy function behavior with unimodular lattices.

A universal method to compute secrecy gain from code weight distribution is proposed.

Formally unimodular packings can achieve higher secrecy gain than the best-known unimodular lattices.

Abstract

This paper introduces the family of lattice-like packings, which generalizes lattices, consisting of packings possessing periodicity and geometric uniformity. The subfamily of formally unimodular (lattice-like) packings is further investigated. It can be seen as a generalization of the unimodular and isodual lattices, and the Construction A formally unimodular packings obtained from formally self-dual codes are presented. Recently, lattice coding for the Gaussian wiretap channel has been considered. A measure called secrecy function was proposed to characterize the eavesdropper's probability of correctly decoding. The aim is to determine the global maximum value of the secrecy function, called (strong) secrecy gain. We further apply lattice-like packings to coset coding for the Gaussian wiretap channel and show that the family of formally unimodular packings shares the same secrecy function behavior as unimodular and isodual lattices. We propose a universal approach to determine the secrecy gain of a Construction A formally unimodular packing obtained from a formally self-dual code. From the weight distribution of a code, we provide a necessary condition for a formally self-dual code such that its Construction A formally unimodular packing is secrecy-optimal. Finally, we demonstrate that formally unimodular packings/lattices can achieve higher secrecy gain than the best-known unimodular lattices.