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, 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.

This paper focuses on the encoding and decoding of Construction D' coding lattices that can be used with shaping lattices for power-constrained channels. Two encoding methods and a decoding algorithm for Construction D' lattices are given. A design of quasi-cyclic low-density parity-check (QC-LDPC) codes to form Construction D' lattices is presented. This allows construction of nested lattice codes which are good for coding, good for shaping, and have low-complexity encoding and decoding. Numerical results using$E_{8},\ BW_{16}$and Leech lattices for shaping a Construction D' lattice indicate that the shaping gains 0.65 dB, 0.86 dB and 1.03 dB are preserved, respectively.

This paper studies the shaping gain and the performance-complexity trade-off of convolutional code lattices, lattices that are based on convolutional codes and Construction A. Generator polynomials for convolutional codes which provide the best-found shaping gain are presented, for various code rates, memory orders and lattice dimensions. Results are based on exhaustive searches. The obtained shaping gains usually exceed that of convolutional codes good for coding. Convolutional code lattices with memory order 7 can provide a shaping gain of 1.20 dB (up to a possible 1.53 dB theoretical maximum) at dimension 256, which is higher than the 1.03 dB shaping gain of the Leech lattice. While convolutional code lattices based on rate 1/2 convolutional codes have the best shaping gain for a fixed memory order, we show that using rate 1/3 convolutional codes produces a more favorable performance-complexity trade-off.