Brett Stevens (Carleton University)
Affine planes with ovals for blocks

A beautiful theorem states that the reverse of a line in the Singer Cycle presentation of a projective plane is an oval, a set of n+1 points that intersect with lines in at most two points. This implies that for every Desarguesian projective plane there is a companion plane all of whose blocks are ovals in the first. This fact has been exploited to construct a family of very efficient strength 3 covering arrays. When Charlie Colbourn visited in April of 2017 he mentioned that he and a student had been able to find pairs of affine planes of orders 3, 4 and 8 with this property and used them to build new strength 3 covering arrays; He asked whether this was true in general for affine planes. In this talk I present a construction of pairs of affine planes whose blocks are ovals in the other plane for any order a power of 2. I discuss issues using these to construct covering arrays.