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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.