Some combinatorial aspects of pedal sets

Pedal sets serve as a useful tool for classifying unitals and can act as a foundation for discovering new unitals. In this article, it is proved that the numbers of pedal sets of line types $ (q+1) $ and $ (q, 2^q) $ in a unital $ U $ (embedded in a projective plane of order $ q^2 $) and its dual unital $ U^\perp $ are equal. Details of the statistics on the pedal sets of unitals in planes of order 25 are presented. Some counter-examples are given to disprove one of the previously proposed conjectures. A new unital embedded in one of the planes of order 25 is discovered. Distributions of collinear pedal sets not lying on a special tangent are analyzed. Computational results presented in this study not only address longstanding inquiries but also prompt new questions for future research directions.