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 Type I 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.

翻译：最近，针对高斯窃听信道中应用的格子码的保密友好性，提出了一个基于格子的体积和theta级数的设计准则，称为保密增益。为了分析从形式自对偶的$\mathbb{Z}_4$-线性码（即对称重量枚举器与其对偶码的枚举器一致的码）构造的$\text{A}_4$格子的保密增益，我们提出了新的$\mathbb{Z}_4$-线性码构造方法，这些码关于对称重量枚举器是形式自对偶的。对于偶数长度，通过嵌套二进制码和双循环矩阵构造了形式自对偶的$\mathbb{Z}_4$-线性码。对于奇数长度，提出了一种新颖的构造——从双循环码进行奇数扩展。此外，引入了类型I/II形式自对偶码/单模格子的概念。接着，我们推导了通过Construction~$\text{A}_4$从形式自对偶$\mathbb{Z}_4$-线性码得到的单模格子的theta级数，并描述了一种计算其保密增益的通用方法。研究了从类型I形式自对偶$\mathbb{Z}_4$-线性码通过Construction $\text{A}_4$得到的单模格子的保密增益，同时考虑了偶数和奇数维度的情况。数值结果表明，对于某些参数，Construction $\text{A}_4$格子能够实现比文献中已知最优形式自对偶格子更高的保密增益。