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.

翻译：近期，学者提出了一种基于格子体积和theta级数的设计准则——称为保密增益——用于量化高斯窃听信道中所用格子码的保密性能。为研究从形式自对偶$\mathbb{Z}_4$线性码（即其对称权重枚举器与其对偶码的对称权重枚举器一致的码）通过构造$\text{A}_4$所得格子的保密增益，本文提出了若干关于$\mathbb{Z}_4$线性码的新构造方法，这些码关于对称权重枚举器具有形式自对偶性。针对偶数长度，利用嵌套二元码和双循环矩阵构造了形式自对偶的$\mathbb{Z}_4$线性码；针对奇数长度，则提出了一种称为"双循环码奇扩展"的新颖构造方法。此外，引入了I型/II型形式自对偶码/幺模格子的概念。接着，我们推导了从形式自对偶$\mathbb{Z}_4$线性码通过构造$\text{A}_4$所得形式幺模格子的theta级数，并描述了一种确定其保密增益的普适方法。本文系统研究了偶数维与奇数维情形下，从形式自对偶$\mathbb{Z}_4$线性码通过构造$\text{A}_4$所得形式幺模格子的保密增益。数值结果表明，对于某些参数，构造$\text{A}_4$格子可实现高于文献中已知最优形式幺模格子的保密增益。