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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 <inline-formula> <tex-math notation="LaTeX">$C^\star $ </tex-math></inline-formula>. We investigate the geometric uniformity, latticeness, and minimum distance properties of Construction <inline-formula> <tex-math notation="LaTeX">$C^\star $ </tex-math></inline-formula>, and discuss its superior packing efficiency when compared to Construction C.

Multilevel Constructions: Coding, Packing and Geometric Uniformity