On t ‐designs and s ‐resolvable t ‐designs from hyperovals

Hyperovals in projective planes turn out to have a link with t ‐designs. Motivated by an unpublished work of Lonz and Vanstone, we present a construction for t ‐designs and s ‐resolvable t ‐designs from hyperovals in projective planes of order 2 n . We prove that the construction works for t ≤ 5 . In particular, for t = 5 the construction yields a family of 5‐ ( 2 n + 2 , 8 , 70 ( 2 n − 2 − 1 ) ) designs. For t = 4 numerous infinite families of 4‐designs on 2 n + 2 points with block size 2 k can be constructed for any k ≥ 4 . The construction assumes the existence of a 4‐ ( 2 n − 1 + 1 , k , λ ) design, called the indexing design, including the complete 4‐ ( 2 n − 1 + 1 , k , (2 n − 1 − 3 k − 4 ) ) design. Moreover, we prove that if the indexing design is s ‐resolvable, then so is the constructed design. As a result, many of the constructed designs are s ‐resolvable for s = 2 , 3 . We include a short discussion on the simplicity or non‐simplicity of the designs from hyperovals.