Cubic Shaping of Lattice Constellations from Multi-Level Constructions from Codes

Lattice codes play an important role in wireless communication and are closely related to linear codes. Multi-level constructions of complex lattices from codes are known to produce lattice codes with desirable parameters and efficient encoding and decoding of information bits. However, their constellation usually involves superfluous elements that need to be mapped to a representative within the same coset to reduce average transmission power. One such elegant shaping function is a componentwise modulo, which is known to produce a cubic shaping for Barnes–Wall lattices. In this paper, we generalize this result to lattices over quadratic rings of integers, thus encompassing constructions from p-ary codes, where p is a prime number. We identify all bases that permit cubic modulo shaping. This provides useful insights into practical encoding and decoding of lattice codes from multi-level constructions.