8 research outputs found
New necessary conditions for (negative) Latin square type partial difference sets in abelian groups
Partial difference sets (for short, PDSs) with parameters (,
, , ) are called
Latin square type (respectively negative Latin square type) PDSs if
(respectively ). In this paper, we will give
restrictions on the parameter of a (negative) Latin square type partial
difference set in an abelian group of non-prime power order. As far as we know
no previous general restrictions on were known. Our restrictions are
particularly useful when is much larger than . As an application, we
show that if there exists an abelian negative Latin square type PDS with
parameter set , , a prime number and is an odd
positive integer, then there are at most three possible values for . For two
of these three values, J. Polhill gave constructions in 2009
Some results on partial difference sets and partial geometries
This thesis shows results on 3 different problems involving partial difference sets (PDS) in abelian groups, and uses PDS to study partial geometries with an abelian Singer group. First, the last two undetermined cases of PDS on abelian groups with k ≤ 100, both of order 216, were shown not to exist. Second, new parameter bounds for k and ∆ were found for PDS on abelian groups of order p^n , p an odd prime, n odd. A parameter search on p^5 in particular was conducted, and only 5 possible such cases remain for p \u3c 250. Lastly, the existence of rigid type partial geometries with an abelian Singer group are examined; existence is left undetermined for 11 cases with α ≤ 200. This final study led to the determination of nonexistence for an infinite class of cases which impose a negative Latin type PDS
Benson\u27s Theorem for Partial Geometries
In 1970 Clark Benson published a theorem in the Journal of Algebra stating a congruence for generalized quadrangles. Since then this theorem has been expanded to other specific geometries. In this thesis the theorem for partial geometries is extended to develop new divisibility conditions for the existence of a partial geometry in Chapter 2. Then in Chapter 3 the theorem is applied to higher dimensional arcs resulting in parameter restrictions on geometries derived from these structures. In Chapter 4 we look at extending previous work with partial geometries with α = 2 to uncover potential partial geometries with higher values of α. Finally the theorem is extended to strongly regular graphs in Chapter 5. In addition we obtain expressions for the multiplicities of the eigenvalues of matrices related to the adjacency matrices of these graphs. Finally, a four lesson high school level enrichment unit is included to provide students at this level with an introduction to partial geometries, strongly regular graphs, and an opportunity to develop proof skills in this new context
Proper partial geometries with Singer groups and pseudogeometric partial difference sets
10.1016/j.jcta.2007.05.001Journal of Combinatorial Theory. Series A1151147-177JCBT