Maximal nonassociativity via fields.

Designs, Codes and Cryptography 88 (2020), 2521-2530.

We say that (x,y,z)\in Q^3 is an associative triple in a quasigroup (Q,*) if (x*y)*z=x*(y*z). Let a(Q) denote the number of associative triples in Q. It is easy to show that a(Q) >= |Q|, and we call the quasigroup maximally nonassociative if a(Q)=|Q|. It was conjectured that maximally nonassociative quasigroups do not exist when |Q|>1. Drapal and Lisonek recently refuted this conjecture by proving the existence of maximally nonassociative quasigroups for a certain infinite set of orders |Q|. In this paper we prove the existence of maximally nonassociative quasigroups for a much larger set of orders |Q|. Our main tools are finite fields and the Weil bound on quadratic character sums. Unlike in the previous work, our results are to a large extent constructive.