Construction D' Lattices for Power-Constrained Communications

TL;DR

This paper develops nested lattice codes using Construction D' lattices and convolutional code lattices for power-constrained channels, achieving high shaping gains and efficient encoding/decoding methods.

Contribution

It introduces new encoding and decoding methods for Construction D' lattices, designs convolutional and QC-LDPC codes with optimized shaping gains, and evaluates their performance on various shaping lattices.

Findings

Rate 1/3 convolutional codes outperform rate 1/2 in performance-complexity trade-off.

Tail-biting convolutional codes have higher shaping gain than zero-tailed codes.

Constructed QC-LDPC Construction D' lattices achieve significant shaping gains on multiple lattices.

Abstract

Designs and methods for nested lattice codes using Construction D' lattices for coding and convolutional code lattices for shaping are described. Two encoding methods and a decoding algorithm for Construction D' coding lattices that can be used with shaping lattices for power-constrained channels are given. We construct nested lattice codes with good coding properties, a 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' lattices which have efficient encoding and indexing is presented. The resulting QC-LDPC Construction D' lattices are evaluated using four shaping lattices: the $E_8$ lattice, the $BW_{16}$ 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.