<p>Recently, Greenfeld and Tao (2024) disproved the conjecture that translational tilings of a single tile can always be periodic. In another paper (Greenfeld and Tao (2025)), they also showed that if the dimension <i>n</i> is part of the input, the translational tiling for subsets of ℤ^<i>n</i> with one tile is undecidable. These two results are very strong pieces of evidence for the conjecture that translational tiling of ℤ^<i>n</i> with a monotile is undecidable for some fixed <i>n</i>. In this paper, we show that translational tiling of the 3-dimensional space with a set of 5 polycubes is undecidable. By introducing a technique that lifts a set of polycubes and its tiling from the 3-dimensional space to the 4-dimensional space, we manage to show that translational tiling of the 4-dimensional space with a set of 4 tiles is undecidable. This is a step towards the attempt to settle the conjecture of the undecidability of translational tiling of the <i>n</i>-dimensional space with a monotile for some fixed <i>n</i>.</p>