Planar cyclic difference packings.

A planar cyclic difference set of order n is an (n + 1)-subset D of the integers modulo n2+n+1 such that every nonzero residue of the integers modulo n2+n+1 is represented exactly once as a difference between elements of D. These combinatorial designs were first studied by Singer in 1938. In 1992, Wiedemann introduced a more general combinatorial design which he called a cyclic difference cover. A cyclic difference cover is a subset D of the integers modulo v such that every nonzero residue of the integers modulo v is represented at least once as a difference between elements of D. The topic of this thesis is a combinatorial design which is complementary to the design studied by Wiedemann. A planar cyclic difference packing modulo v is a subset D of the integers modulo v such that every nonzero residue of the integers modulo v is represented at most once as a difference between elements of D. This thesis develops various constructions of such designs and for a fixed positive integer v presents upper and lower bounds on Yv , the maximal number of elements that a planar cyclic difference packing modulo v can contain. Using a Maple procedure, Yv is calculated for v≤144 . This thesis also presents results towards proving the 0,1 -Conjecture, that for any fixed v there exists a planar cyclic difference packing modulo v containing Yv elements including 0,1 . Finally, the thesis presents open questions regarding these designs and definitions of more generalized combinatorial designs for future research.Ph.D.MathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/132015/2/9938549.pd