A lattice hypercube is a cube in 4-dimensional space, whose vertices are lattice points. Describe the set of possible side lengths of a lattice hypercube.

**Note**: A **lattice point** is a point whose coordinates are all integers. \( (0,0,1) \) is a lattice point, but \( \left(\frac{1}{2}, \sqrt{2}, -0.2 \right) \) is not a lattice point.

If you are interested in Lattice cubes, try the 3-D version.

For similar problems, you can read my note on Construction.

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