Secrecy Gain of Formally Unimodular Lattices from Codes over the Integers Modulo 4
Maiara F. Bollauf, Hsuan-Yin Lin, Øyvind Ytrehus
TL;DR
The paper analyzes the secrecy performance of Construction $\textnormal{A}_4$ lattices built from formally self-dual $\mathbb{Z}_4$-linear codes for Gaussian wiretap channels. It introduces new fsd code constructions for both even and odd lengths (including odd extensions of double circulant codes), derives theta-series expressions for the resulting lattices, and provides a universal framework to compute secrecy gains via a closed-form in terms of the symmetrized weight enumerator. The authors show, numerically, that certain Construction $\textnormal{A}_4$ lattices achieve higher secrecy gains than the best-known formally unimodular lattices, and they connect secrecy gain to the flatness factor, offering guidance for practical wiretap design. The work also provides upper bounds for Type I formally unimodular lattices and demonstrates the value of Gray-mapped binary codes derived from $\mathbb{Z}_4$ codes, broadening the toolkit for secrecy-enabled lattice design.
Abstract
Recently, a design criterion depending on a lattice's volume and theta series, called the secrecy gain, was proposed to quantify the secrecy-goodness of the applied lattice code for the Gaussian wiretap channel. To address the secrecy gain of Construction $\text{A}_4$ lattices from formally self-dual $\mathbb{Z}_4$-linear codes, i.e., codes for which the symmetrized weight enumerator (swe) coincides with the swe of its dual, we present new constructions of $\mathbb{Z}_4$-linear codes which are formally self-dual with respect to the swe. For even lengths, formally self-dual $\mathbb{Z}_4$-linear codes are constructed from nested binary codes and double circulant matrices. For odd lengths, a novel construction called odd extension from double circulant codes is proposed. Moreover, the concepts of Type I/II formally self-dual codes/unimodular lattices are introduced. Next, we derive the theta series of the formally unimodular lattices obtained by Construction $\text{A}_4$ from formally self-dual $\mathbb{Z}_4$-linear codes and describe a universal approach to determine their secrecy gains. The secrecy gain of Construction $\text{A}_4$ formally unimodular lattices obtained from formally self-dual $\mathbb{Z}_4$-linear codes is investigated, both for even and odd dimensions. Numerical evidence shows that for some parameters, Construction $\text{A}_4$ lattices can achieve a higher secrecy gain than the best-known formally unimodular lattices from the literature. Results concerning the flatness factor, another security criterion widely considered in the Gaussian wiretap channel, are also discussed.