Multilevel Constructions: Coding, Packing and Geometric Uniformity

Lattice and special nonlattice multilevel constellations constructed from binary codes, such as Constructions A, C, and D, have relevant applications in Mathematics (sphere packing) and in Communication (multi-stage decoding and efficient vector quantization). In this work, we explore some properties of Construction C, in particular its geometric uniformity. We then propose a new multilevel construction, inspired by bit interleaved coded modulation (BICM), that we call Construction
$C^\star $
. We investigate the geometric uniformity, latticeness, and minimum distance properties of Construction
$C^\star $
, and discuss its superior packing efficiency when compared to Construction C.