Publication at Faculty of Mathematics and Physics |

2023

Let q be an odd prime power and suppose that a,bELEMENT OFFq are such that ab and (1-a)(1-b) are nonzero squares. Let Qa,b=(Fq,ASTERISK OPERATOR) be the quasigroup in which the operation is defined by uASTERISK OPERATORv=u+a(v-u) if v-u is a square, and uASTERISK OPERATORv=u+b(v-u) if v-u is a nonsquare.

This quasigroup is called maximally nonassociative if it satisfies xASTERISK OPERATOR(yASTERISK OPERATORz)=(xASTERISK OPERATORy)ASTERISK OPERATORzLEFT RIGHT DOUBLE ARROWx=y=z. Denote by σ(q) the number of (a,b) for which Qa,b is maximally nonassociative.

We show that there exist constants αALMOST EQUAL TO0.02908 and βALMOST EQUAL TO0.01259 such that if qIDENTICAL TO1mod4, then limσ(q)/q2=α, and if qIDENTICAL TO3mod4, then limσ(q)/q2=β.