Nonextendible Latin Cuboids

We show that for all integers m > 4 there exists a 2m × 2m ×m latin cuboid that cannot be completed to a 2m × 2m × 2m latin cube. We also show that for all even m / ∈ {2, 6} there exists a (2m−1) × (2m−1) × (m−1) latin cuboid that cannot be extended to any (2m−1)× (2m−1)×m latin cuboid.