A characterization of the circularity of certain balanced incomplete block designs.

Abstract

When defining a structure to fulfill a set of axioms that are similar to those prescribed by Euclid, one must select a set of points and then define what is meant by a line and what is meant by a circle. When properly defined these labels will have properties which are similar to their counterparts in the (complex) plane, the lines and circles which Euclid undoubtedly had in mind. In this manner, the geometer may employ his intuition from the complex plane to prove theorems about other systems. Most "finite geometries" have clearly defined notions of points and lines but fail to define circles. The two notable exceptions are the circles in a finite affine plane and the circles in a Mobius plane. Using the geometry of Euclid as motivation, we strive to develop structures with both lines and circles. The only successful example other than the complex plane is the affine plane over a finite field, where all of Euclid's geometry holds except for any assertions involving order or continuity. To complement the prolific work concerning finite geometries and their lines, we provide a general definition of a circle, or more correctly, of a collection of circles and present some preliminary results concerning the construction of such structures. Our definition includes the circles of an affine plane over a finite field and the circles in a Mobius plane as special cases. We develop a necessary and sufficient condition for circularity, present computational techniques for determining circularity and give varying constructions. We devote a chapter to the use of circular designs in coding theory. It is proven that these structures are not useful in the theory of error-correcting codes, since more efficient codes are known, for example the Reed-Muller codes. However, the theory developed in the earlier chapters does have applications to Cryptology. We present five encryption methods utilizing circular structures.