Concatenated codes are examples of compound constructions, as they are obtained by combining two codes—an inner code and an outer code—with a certain relationship between their parameters. This chapter presents another compound construction, now combining an (inner) code C over some alphabet F with an undirected graph G = (V, E). In the resulting construction, which we refer to as a graph code and denote by (G, C), the degrees of all the vertices in G need to be equal to the length of C, and the code (G, C) consists of all the words of length ∣E∣ over F in which certain sub-words, whose locations are defined by G, belong to C. The main result to be obtained in this chapter is that there exist explicit constructions of graph codes that can be decoded in linear-time complexity, such that the code rate is bounded away from zero, and so is the fraction of symbols that are allowed to be in error.

We start this chapter by reviewing several concepts from graph theory. We then focus on regular graphs, i.e., graphs in which all vertices have the same degree. We will be interested in the expansion properties of such graphs; namely, how the number of outgoing edges from a given set of vertices depends on the size of this set.