Small complete arcs in \(PG(2,p)\) (Q1284207)

A \(k\)-arc in a projective plane is a set of \(k\) points no three of which are collinear. A \(k\)-arc is called complete if it cannot be extended to a \(k+1\)-arc. Let \(p\) be a prime. Hadnagy constructs a family of complete \(k\)-arcs in the projective plane \(PG(2,p)\) for all values of \(k\) in the range \[ \left[ 2.46\cdot \ln p\cdot p ^{3\over 4}\right] \leq k \leq \left[ {p+7 \over 3}\right] + 1. \] This result supplements results by \textit{J. F. Voloch} [Eur. J. Comb. 8, 453-456 (1987; Zbl 0645.51010)] and \textit{T. Szönyi} [Combinatorics, Proc. 7th Hung. Colloq., Eger/Hung. 1987, Colloq. Math. Soc. János Bolyai 52, 499-508 (1988; Zbl 0685.51005)] obtaining the following corollary. Let \(p\) be a prime. Then, for any integer \(k\) satisfying \[ \left[ 2.46\cdot \ln p\cdot p ^{3\over 4}\right] \leq k \leq (\sqrt{p}+1)^2 /2, \] there exists a complete \(k\)-arc in \(PG(2,p)\). The construction of Hadnagy starts with a set of points on a rational cubic curve that form an arc. Complete arcs are obtained by adding some extra points. The proofs focus on finding maximal 3-independent subsets of the additive group modulo \(p\).