  Sampling from the lattice Gaussian distribution has emerged as an important
problem in coding, decoding and cryptography. In this paper, lattice reduction
technique is adopted to Gibbs sampler for lattice Gaussian sampling. Firstly,
with respect to lattice Gaussian distribution, the convergence rate of
systematic scan Gibbs sampling is derived and we show it is characterized by
the Hirschfeld-Gebelein-R\'enyi (HGR) maximal correlation among the
multivariate of being sampled. Therefore, lattice reduction is applied to
formulate an equivalent lattice Gaussian distribution but with less correlated
multivariate, which leads to a better Markov mixing due to the enhanced
convergence rate. Then, we extend the proposed lattice-reduction-aided Gibbs
sampling to lattice decoding, where the choice of the standard deviation for
the sampling is fully investigated. A customized solution that suits for each
specific decoding case by Euclidean distance is given, thus resulting in a
better trade-off between Markov mixing and sampler decoding. Moreover, based on
it, a startup mechanism is also proposed for Gibbs sampler decoding, where
decoding complexity can be reduced without performance loss. Simulation results
based on large-scale MIMO detection are presented to confirm the performance
gain and complexity reduction.
