Geometrically-Shaped Constellation for Visible Light Communications at Short Blocklength

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.