Construction D’ Lattices for Power-Constrained Communications

Abstract

Designs and methods for nested lattice codes using Construction D&#x2019; lattices for coding and convolutional code lattices for shaping are described. Two encoding methods and a decoding algorithm for Construction D&#x2019; coding lattices that can be used with shaping lattices for power-constrained channels are given. We construct nested lattice codes with good coding properties, high shaping gain, and low-complexity encoding and decoding. Convolutional code generator polynomials for Construction A lattices with the greatest shaping gain are given, as a result of an extensive search. It is shown that rate 1/3 convolutional codes provide a more favorable performance-complexity trade-off than rate 1/2 convolutional codes. Tail-biting convolutional codes have higher shaping gain than that of zero-tailed convolutional codes. A design for quasi-cyclic low-density parity-check (QC-LDPC) codes to form Construction D&#x2019; lattices which have efficient encoding and indexing is presented. The resulting QC-LDPC Construction D&#x2019; lattices are evaluated using four shaping lattices: the <inline-formula> <tex-math notation="LaTeX">$E_{8}$ </tex-math></inline-formula> lattice, the <inline-formula> <tex-math notation="LaTeX">$BW_{16}$ </tex-math></inline-formula> lattice, the Leech lattice and our best-found convolutional code lattice, showing a shaping gain of approximately 0.65 dB, 0.86 dB, 1.03 dB and 1.25 dB at dimension 2304.