  Sequential Latin hypercube designs have recently received great attention for
computer experiments. Much of the work has been restricted to invariant spaces.
The related systematic construction methods are inflexible while algorithmic
methods are ineffective for large designs. For such designs in space
contraction, systematic construction methods have not been investigated yet.
This paper proposes a new method for constructing sequential Latin hypercube
designs via good lattice point sets in a variety of experimental spaces. These
designs are called sequential good lattice point sets. Moreover, we provide
fast and efficient approaches for identifying the (nearly) optimal sequential
good lattice point sets under a given criterion. Combining with the linear
level permutation technique, we obtain a class of asymptotically optimal
sequential Latin hypercube designs in invariant spaces where the $L_1$-distance
in each stage is either optimal or asymptotically optimal. Numerical results
demonstrate that the sequential good lattice point set has a better
space-filling property than the existing sequential Latin hypercube designs in
the invariant space. It is also shown that the sequential good lattice point
sets have less computational complexity and more adaptability.
