Subsets of $${\mathbb {P}}^4$$ with no four points on a plane

Abstract We describe a new construction of a subset of $${\mathbb {P}}^4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> </mml:math> with no four points on a plane over any finite field of order q in which 3 is not a square. This set has size $$2q+1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>q</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , is maximal with respect to inclusion, and is the largest known such set.