Designs from cyclotomy

In this note we use the theory of cyclotomy to help us construct initial blccks from which we can develop balanced and partially balanced incomplete block designs. Our main construction method, using unions of cyclotomic classes, gives us upper bounds on m, the number of associate classes of the design, but not exact values for m; we discuss the possible values of m and the circumstances under which m = 1, so that the design is in fact balanced.