On bisecants of Rédei type blocking sets and applications (Q1747997)

If \(q > 13\), \(q\) is a power of the prime \(p\), then there is only one known minimal blocking set of size \(3(q+1)/2\) in \(\operatorname{PG}(2, q)\), the so-called projective triangle. This blocking set is of Rédei type and it has \(3(q-1)/2\) bisecants, which have a very strict structure. In this paper polynomial techniques are used to derive structural results on Rédei type blocking sets from information on their bisecants. Blokhuis characterized semiovals in \(\operatorname{PG}(2, q)\) of size \(q-1+a\), \(a>2\), meeting each line in 0, 1, 2, or \(a\) points. He also proved that there is no semioval of size \(q+2\) in \(\operatorname{PG}(2, q)\), \(q>7\). In the reviewed paper the Blokhuis' characterization is refined to obtain new structural results about semiovals of size \(q-1+a\) containing \(a\) collinear points. As an application, the non-existence of semiovals of size \(q+3\) in \(\operatorname{PG}(2, q)\) is proved, \(5<q\) odd when \(p \neq 3\). In [J. Comb. Des. 22, No. 10, 435--451 (2014; Zbl 1310.51004)], \textit{P. Balister} et al. investigated the minimum number of odd-secants of an \(n\)-set in \(\operatorname{PG}(2, q)\), \(q\) odd. They studied in detail the case of \(n=q+2\). Their lower bound is improved in the reviewed paper and a related open question from \textit{P. Vandendriessche} [Des. Codes Cryptography 75, No. 3, 453--463 (2015; Zbl 1312.05029)] is settled.