Embedding in MDS codes and Latin cubes

TL;DR

This paper proves that any code with a given distance and length can be embedded into an MDS code with a larger alphabet, also enabling embeddings of Latin cubes and quasigroups, thus linking coding theory and combinatorial designs.

Contribution

It establishes a general embedding theorem for codes into MDS codes and applies it to Latin cubes and quasigroups, extending the understanding of their structural relationships.

Findings

Any code with distance ρ and length d can be embedded into an MDS code with the same parameters.

Embeddings of systems of partial mutually orthogonal Latin cubes are possible.

The results connect coding theory with combinatorial structures like Latin cubes and quasigroups.

Abstract

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.