In this paper, we present a general framework of designing geometrically shaped constellations for short-packet visible light communications with a peak- and an average-intensity constraints. By leveraging tools from large deviation theory, we first characterize the second-order asymptotics of the optimal constellation shaping region under aforementioned intensity constraints, which serves as a good performance measure for the best geometric shaping in finite blocklength. To further incorporate a sufficiently large coding gain and a nearly-maximum shaping gain, we construct multidimensional constellations by the nested structure of Construction B lattices, where the constellation shaping is implemented by controlling the boundary of the embedded sublattice, i.e., a strategy called coarsely shaping and finely coding. Fast algorithms for constellation mapping and demodulation are presented as well. As an illustrative example, we present an energy-efficient $24$-dimensional constellation design based on the Leech lattice, whose superiority over existing constellation designs is verified by numerical results.

翻译：本文提出了一种在峰值与平均强度约束下，为短包可见光通信设计几何成形星座的通用框架。首先，利用大偏差理论工具，刻画了在上述强度约束下最优星座成形区域的二阶渐近特性，该特性可作为有限块长下最佳几何成形性能的优良度量。为进一步融合足够大的编码增益与近乎最大的成形增益，我们通过构造B格（Construction B lattices）的嵌套结构构建了多维星座，其中星座成形通过控制嵌入子格的边界实现——即称为“粗成形与细编码”的策略。同时给出了星座映射与解调的快速算法。作为示例，我们基于李奇格（Leech lattice）提出了一种能量高效的24维星座设计，数值结果验证了其相较于现有星座设计的优越性。