Abstract We present some new results on incorporating quasi-Monte Carlo rules into Markov chain Monte Carlo. First, we present some new constructions of points, fully equidistributed LFSRs, which are small enough that the entire point set can be used in a Monte Carlo calculation. Second, we introduce some antithetic and round trip sampling constructions and show that they preserve the completely uniformly distributed property necessary for QMC in MCMC. Finally, we also give some new empirical results. We see large improvements in sampling some GARCH and stochastic volatility models.