In this note we prove the non-existence of two types of partial difference sets in Abelian groups of order 216. This finalizes the classification of parameters for which a partial difference set of size at most 100 exists in an Abelian group.

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A partial geometry S admitting an abelian Singer group G is called of rigid type if all lines of S have a trivial stabilizer in G. In this paper, we show that if a partial geometry of rigid type has fewer than 1000000 points it must be the Van Lint-Schrijver geometry or be a hypothetical geometry with 1024 or 4096 or 194481 points, which provides evidence that partial geometries of rigid type are very rare. Along the way we also exclude an infinite set of parameters that originally seemed very promising for the construction of partial geometries of rigid type (as it contains the Van Lint-Schrijver parameters as its smallest case and one of the other cases we cannot exclude as the second member of this parameter family). We end the paper with a conjecture on this type of geometries.

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