Design theory
Sub-difference sets of Hadamard difference sets
Journal of Combinatorial Theory Series A
On subsets of partial difference sets
Selected papers of the 13th British Combinatorial Conference on British combinatorial conference
A survey of partial difference sets
Designs, Codes and Cryptography
Designs, Graphs, Codes, and Their Links
Designs, Graphs, Codes, and Their Links
Elation and translation semipartial geometries
Journal of Combinatorial Theory Series A
Journal of Algebraic Combinatorics: An International Journal
Designs, Codes and Cryptography
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A partial geometry admitting a Singer group G is equivalent to a partial difference set in G admitting a certain decomposition into cosets of line stabilizers. We develop methods for the classification of these objects, in particular, for the case of abelian Singer groups. As an application, we show that a proper partial geometry @P=pg(s+1,t+1,2) with an abelian Singer group G can only exist if t=2(s+2) and G is an elementary abelian 3-group of order (s+1)^3 or @P is the Van Lint-Schrijver partial geometry. As part of the proof, we show that the Diophantine equation (3^m-1)/2=(2^r^w-1)/(2^r-1) has no solutions in integers m,r=1, w=2, settling a case of Goormaghtigh's equation.