  Block-fading channel (BF) is a useful model for various wireless
communication channels in both indoor and outdoor environments. The design of
lattices for BF channels offers a challenging problem, which differs greatly
from its counterparts like AWGN channels. Recently, the original binary
Construction A for lattices, due to Forney, has been generalized to a lattice
construction from totally real and complex multiplication (CM) fields. This
generalized algebraic Construction A of lattices provides signal space
diversity, intrinsically, which is the main requirement for the signal sets
designed for fading channels. In this paper, we construct full-diversity
algebraic lattices for BF channels using Construction A over totally real
number fields. We propose two new decoding methods for these lattices which
have complexity that grows linearly in the dimension of the lattice. The first
decoder is proposed for generalized Construction A lattices with a binary LDPC
code as underlying code. This decoding method contains iterative and
non-iterative phases. In order to implement the iterative phase, we propose the
definition of a parity-check matrix and Tanner graph for Construction A
lattices. We also prove that using an underlying LDPC code that achieves the
outage probability limit over one-BF channel, the constructed algebraic LDPC
lattices together with the proposed decoding method admit diversity order n.
Then, we modify the proposed algorithm by removing its iterative phase which
enables full-diversity practical decoding of all generalized Construction A
lattices without any assumption about their underlying code. We provide some
instances showing that algebraic Construction A lattices obtained from binary
codes outperform the ones based on non-binary codes in BF channels. We
generalize algebraic Construction A lattices over a wider family of number
fields namely monogenic number fields.
