  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.
