3 - Coding and Combinatorics

Abstract

This chapter describes coding and combinatorics. It presents the relationship between coding theory and certain aspects of experimental design and matroid theory. With every linear code, there is an associated combinatorial structure called a matroid. Many of the investigations concerning designs and codes are in fact investigations of the properties of the matroids of linear codes. Chain groups are studied because of the fact that structures have properties that are reflected in certain of the chain groups. If there is a linear graph whose edge set is 5, its polygon matroid has certain properties that must be reflected in certain chain groups that share the same matroid, which allows one to determine whether or not a given matroid is cographic, that is, the polygon matroid of a linear graph. The theory of matroids is a rich and beautiful area of combinatorial mathematics, encompassing the area of finite geometries as well. The girth of a graph is the number of edges contained in the smallest polygon of the graph.