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.

翻译：本文引入了拟格填充族，该类填充作为格的一般化形式，兼具周期性与几何均匀性。进一步研究了形式无模拟格填充子族，该子族可视为无模格与等对偶格的推广，并给出了基于形式自对偶码的构造A型形式无模拟格填充。针对近期高斯窃听信道中格编码的研究，本文引入名为保密函数的度量来表征窃听者正确解码的概率，其核心目标是确定该函数的全局最大值（即强保密增益）。我们进一步将拟格填充应用于高斯窃听信道的陪集编码，证明形式无模拟格填充族与无模格、等对偶格具有相同的保密函数特性。我们提出了一种通用方法，用于确定基于形式自对偶码的构造A型形式无模拟格填充的保密增益。通过码的权重分布，给出了形式自对偶码满足其构造A型形式无模拟格填充达到最优保密性的必要条件。最后，我们证明形式无模拟格填充/格能够比已知最优的无模格实现更高的保密增益。