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 d $d$ and length n $n$ can be embedded into an maximum distance separable (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 $n$-ary quasigroups.