Difference sets in a finite group

The (16, 16, 2) designs

“…The (16,6,2) designs served and indeed an "Hamada Conjecture" is verified via elementary arguments in our Theorem I.…”

“…The three non-isomorphic designs with parameters (16,6,2) have been investigated from several points of view [3,6,II,12]. The vantage point of algebraic coding theory makes quite explicit certain heretofore unknown relations among them.…”

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“…An invariant for partial relative difference sets. The following lemma is a generalisation of a lemma of Bruck [Bru55] and the proof is very similar to the original one.…”

On polar ovals in cyclic projective planes

“…We shall deduce this from the result of Ott [21] with its generalization by Ho [13], and the classic characterization of the multiplier group by Bruck [4], using the polarity β provided by Theorem 3.11. Furthermore, we shall use the fact that in a Desarguesian plane of odd order q, a classical unital can never contain a conic.…”

“…In this paper, we isolate a mild necessary and sufficient condition, configurational in nature, for an APP of odd order to be Desarguesian. Essential to our approach is the result of Ott [21] together with its generalization to the abelian case by Ho [13], and the classic characterization of the multiplier group by Bruck [4], which combine to give the conclusion that an APP π with abelian Singer group G is either Desarguesian or G is normal in the full automorphism group Aut(π). We shall construct a polarity on an odd order APP satisfying our condition so that its composition with the Hall polarity associated to an APP defines a collineation which is not in the normalizer of G in Aut(π).…”