  Most multi-dimensional (more than two dimensions) lattice partitions only
form additive quotient groups and lack multiplication operations. This prevents
us from constructing lattice codes based on multi-dimensional lattice
partitions directly from non-binary linear codes over finite fields. In this
paper, we design lattice codes from Construction A lattices where the
underlying linear codes are non-binary irregular repeat-accumulate (IRA) codes.
Most importantly, our codes are based on multi-dimensional lattice partitions
with finite constellations. We propose a novel encoding structure that adds
randomly generated lattice sequences to the encoder's messages, instead of
multiplying lattice sequences to the encoder's messages. We prove that our
approach can ensure that the decoder's messages exhibit permutation-invariance
and symmetry properties. With these two properties, the densities of the
messages in the iterative decoder can be modeled by Gaussian distributions
described by a single parameter. With Gaussian approximation, extrinsic
information transfer (EXIT) charts for our multi-dimensional IRA lattice codes
are developed and used for analyzing the convergence behavior and optimizing
the decoding thresholds. Simulation results show that our codes can approach
the unrestricted Shannon limit within 0.46 dB and outperform the previously
designed lattice codes with two-dimensional lattice partitions and existing
lattice coding schemes for large codeword length.
