Cyclic Difference Sets, a summer 2007 project directed by Dr. Ken Smith

A difference set D in an abelian group G with “intersection parameter lambda” has the property that every nonzero element of the group G may be written in exactly lambda ways as a difference of elements of D. For example, in the additive group of integers modulo 7, the set {1, 2, 4} is a difference set with lambda = 1. The list 1-2, 1-4, 2-1, 2-4, 4-1 and 4-2 covers the nonzero elements 1,2,3,4,5,6 exactly once.

Difference sets give rise to symmetric designs with a “sharply transitive” automorphism group.

One may build a difference set in an abelian group by examining idempotents of the associated group ring. Using ring theory, group characters and algebraic number theory, we can create a list of conditions that must be satisfied by a difference set. These conditions allow us to either construct a difference set or rule them out.

In summer 2007 we will focus primarily on open parameters for difference sets in cyclic groups.

Some earlier summer projects explored difference sets in a variety of types of groups. Reports from earlier projects (2002-2006) are available at links on the main REU webpage.