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:最近,根据拉蒂斯的音量和称为“保密增益”的系列,提出了一个设计标准,以量化高斯窃听频道应用的拉蒂斯码的保密性。为了解决建筑公司($\text{A<unk> 4$4$)的保密性收益问题,从正式的自成双形的硬体代码($\mathbb4$4$-线性代码)中,即由双倍的硬体代号(swe)与其双倍的双倍的硬体代码(swe)相匹配的代码。此外,我们提出了新的“$mathbbb4$4线性代码”的构建,这些代码对于高斯调频道而言是自成一体的自成一体的。对于即使长度,也正式的自成一体的自成体系代码($_ma_ma_d_ad) 4,我们正式地从一个自成正态的硬体代码($_ma_maxxxxxxxxxxxal_al_deal) 开始,我们从一个自成正成正成正正的自建的自建的自建的代码(al__xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx</s>