Embedding in MDS codes and Latin cubes

An embedding of a code is a mapping that preserves distances between codewords. We prove that any code with code distance $\rho$ and length $d$ can be embedded into an MDS code with the same code distance and length but under a larger alphabet. As a corollary we obtain embeddings of systems of partial mutually orthogonal Latin cubes and $n$-ary quasigroups.