  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.
