On the Structure of Semiovals of Small Size

Abstract

The classification of all semiovals and blocking semiovals in PG(2, 8) is determined. Also, new theoretical results on the structure of semiovals containing a (q−1)‐secant and some nonexistence results are presented.

A minimum blocking semioval in PG(2, 9)

“…In Desarguesian projective planes, Héger and Takáts [7, Corollary 33] provide the asymptotically better lower bound of 9 4 q−3 for a blocking semioval in P G (2, q). However, in small order planes both of these lower bounds are known not to be sharp; in P G (2,8) these bound admit the possibility of a blocking semioval with 18 points (Dover bound; Héger and Takáts bound is 15), but per Dover [5,Theorem 4.5] the smallest blocking semioval in P G (2,8) has 19 points.…”

“…In [10, Theorems 3.2, 4.2, and 5.2] Nakagawa and Suetake provide examples of blocking semiovals of size 21 in the three non-Desarguesian planes of order nine, which are necessarily minimal. However each of the size 21 blocking semiovals in these planes has an 8-secant, and Suetake [11,Theorem 6.2] showed that P G (2,9) has no blocking semioval of size 21 with an 8-secant, so there is no hope of extending their non-Desarguesian constructions. Suetake [11, Theorem 6.2] does provide a construction of a blocking semioval of size 22 in P G (2,9).…”

“…To summarize, the minimum size of a blocking semioval in every projective plane of order less than 11 is known, except for P G (2,9). In P G (2,9), the minimum size of a blocking semioval is either 21 or 22, and if a blocking semioval of size 21 exists it does not have an 8-secant.…”

“…In P G (2,9), the minimum size of a blocking semioval is either 21 or 22, and if a blocking semioval of size 21 exists it does not have an 8-secant. In the next section, we exhibit a blocking semioval of size 21 in P G (2,9).…”

“…The minimum size of a blocking semioval is currently known in all projective planes of order <11, with the exception of P G (2,9). In this note we show by demonstration of an example that the smallest blocking semioval in P G(2, 9) has size 21 and investigate some properties of this set.…”