Design theory
A survey of partial difference sets
Designs, Codes and Cryptography
Arcs and Ovals from Abelian Groups
Designs, Codes and Cryptography
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A failed symmetric design is a symmetric incidence structure in which any two points are joined by exactly one or λ blocks, and dually. We construct a failed biplane with block size k whenever k - 1 is a prime power and a failed triplane with block size k whenever k + 1 is a prime power congruent to 1 modulo 3. In fact, our examples admit cyclic Singer groups and thus belong to failed difference sets. Finally, we also exhibit a related infinite series of partially symmetric designs with indices λ1 = 2 and λ2 = 3.