3 - Coding and Combinatorics

Abstract

Being concerned with problems of a finite nature, coding theory itself can be considered a part of combinatorial theory. For example, certain coding problems themselves were treated under the title of packing problems before the concept of coding theory arose. This chapter discusses the relationship between coding theory and certain aspects of experimental design and matroid theory. A matroid is graphic if it is the bond matroid of a graph; it is cographic if it is the polygon matroid of a graph. The proposition that every maximum-distance-separable linear binary code can be viewed as the cycle space of a linear graph follows from the fact that all maximum-distance-separable binary codes are of a trivial nature. The fact that the code is maximum distance separable if and only if the belts of its matroid form a full combinatorial design proves that any binary maximum-distance-separable code is trivial.