Two infinite families of failed symmetric designs Original Research Article

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.