2 - Combinatorial Constructions and Coding

Abstract

This chapter presents a reasonably self-contained account of certain combinatorial structures, their interrelationships, and their relationships to codes. The study of geometries is the study of a set of axioms concerning abstract quantities called points, lines, and flats. The axioms are usually framed to satisfy some geometric notions, and there is a considerable interplay between the algebraic and geometric properties of such systems. The chapter presents the definition of finite projective and Euclidean geometries and explains their elementary properties. A finite projective geometry consists of a finite set of elements called points and a family of subsets called lines, which satisfy a number of axioms. The development of Euclidean geometry codes proceeds along lines similar to those of projective geometry codes. The arrangement of the elements of some set into blocks or subsets with certain properties is a problem that attracts much attention, both in mathematics and the applied sciences. Many of the applications lie in the theory of experimental design, but the problems encountered in the study of such systems are of considerable mathematical complexity.