A superlinear improvement on line-free sets in Fp3

Abstract

Building on an earlier result of the author together with Elsholtz, Führer, Füredi, Pach, Simon and Velich, we present an improved construction for a line-free set in 𝔽 3 𝑝 , having size at least ( 𝑝 − 1 ) 3 + 1 8 ⁢ 𝑝 3 / 2 − 𝑂 ⁡ ( 𝑝 ) as 𝑝 → ∞ . This results in a superlinear-term improvement over the standard hypercube construction { 0 , 1 , … , 𝑝 − 2 } 3 . Via complementation, a line-free set in 𝔽 3 𝑝 corresponds to a 2-blocking set in the affine geometry A G ⁡ ( 3 , 𝑝 ) , hence we also obtain an upper bound of 3 ⁢ 𝑝 2 − 1 8 ⁢ 𝑝 3 / 2 + 𝑂 ⁡ ( 𝑝 ) on the smallest size of such a 2-blocking set.