Negative Latin square type partial difference sets and amorphic association schemes with Galois rings

2009 ◽  
Vol 17 (3) ◽  
pp. 266-282 ◽  
Author(s):  
John Polhill
Keyword(s):  
Latin Square ◽  
Difference Sets ◽  
Association Schemes ◽  
Galois Rings ◽  
Partial Difference ◽  
Partial Difference Sets

scholarly journals Switching of Edges in Strongly Regular Graphs I: A Family of Partial Difference Sets on 100 Vertices

10.37236/1710 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
L. K. Jørgensen ◽  
M. Klin
Keyword(s):  
Automorphism Groups ◽  
Difference Sets ◽  
Association Schemes ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Partial Difference ◽  
Galois Correspondence ◽  
Partial Difference Sets ◽  
Strongly Regular ◽  
Open Question

We present 15 new partial difference sets over 4 non-abelian groups of order 100 and 2 new strongly regular graphs with intransitive automorphism groups. The strongly regular graphs and corresponding partial difference sets have the following parameters: (100,22,0,6), (100,36,14,12), (100,45,20,20), (100,44,18,20). The existence of strongly regular graphs with the latter set of parameters was an open question. Our method is based on combination of Galois correspondence between permutation groups and association schemes, classical Seidel's switching of edges and essential use of computer algebra packages. As a by-product, a few new amorphic association schemes with 3 classes on 100 points are discovered.

scholarly journals Generalizations of Partial Difference Sets from Cyclotomy to Nonelementary Abelian $p$-Groups

10.37236/849 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
John Polhill
Keyword(s):  
Abelian Groups ◽  
Latin Square ◽  
Difference Sets ◽  
Difference Set ◽  
Regular Graphs ◽  
Partial Difference Set ◽  
Partial Difference ◽  
Partial Difference Sets ◽  
Similar Product ◽  
Strongly Regular

A partial difference set having parameters $(n^2, r(n-1), n+r^2-3r,r^2-r)$ is called a Latin square type partial difference set, while a partial difference set having parameters $(n^2, r(n+1), -n+r^2+3r,r^2+r)$ is called a negative Latin square type partial difference set. In this paper, we generalize well-known negative Latin square type partial difference sets derived from the theory of cyclotomy. We use the partial difference sets in elementary abelian groups to generate analogous partial difference sets in nonelementary abelian groups of the form $(Z_p)^{4s} \times (Z_{p^s})^4$. It is believed that this is the first construction of negative Latin square type partial difference sets in nonelementary abelian $p$-groups where the $p$ can be any prime number. We also give a generalization of subsets of Type Q, partial difference sets consisting of one fourth of the nonidentity elements from the group, to nonelementary abelian groups. Finally, we give a similar product construction of negative Latin square type partial difference sets in the additive groups of $(F_q)^{4t+2}$ for an integer $t \geq 1$. This construction results in some new parameters of strongly regular graphs.

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