#### Construction D' Lattices for Power-Constrained Communications

##### Fan Zhou, Brian M. Kurkoski

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 which are good for coding, good for shaping, and have low-complexity encoding and decoding. An indexing method for nested lattice codes is modified to avoid an integer overflow problem at high dimension. Convolutional code generator polynomials for Construction A lattices with the greatest shaping gain are given, the 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. For a given dimension, tail-biting convolutional codes have higher shaping gain than that of zero-tailed convolutional codes. A design for quasi-cyclic low-density parity-check (LDPC) codes to form Construction D' lattices is presented, where their parity-check matrices can be easily triangularized, thus enabling efficient encoding and indexing. The resulting 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.

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