Graph Theoretic Methods in Coding Theory

This paper is a tutorial on the application of graph theoretic techniques in classical coding theory. A fundamental problem in coding theory is to determine the maximum size of a code satisfying a given minimum Hamming distance. This problem is thought to be extremely hard and still not completely solved. In addition to a number of closed form expressions for special cases and some numerical results, several relevant bounds have been derived over the years. We show here that many of these bounds can be derived using special properties of certain graphs. For instance, the well-known Gilbert-Varshamov bound can be easily proven using a general bound on the chromatic number of graphs. Furthermore, both the Hamming and Singleton bounds can be derived as an application of a property relating the clique and independence numbers of vertex transitive graphs.