Explicit constructions of Diophantine tuples over finite fields

A Diophantine m-tuple over a finite field Fq is a set ₁,…, aₘ\ of m distinct elements in Fq* such that aᵢaⱼ+1 is a square in Fq whenever i≠ j. In this paper, we study $M(q)$, the maximum size of a Diophantine tuple over Fq, assuming the characteristic of Fq is fixed and q → ∞. By explicit constructions, we improve the lower bound on $M(q)$. In particular, this improves a recent result of Dujella and Kazalicki by a multiplicative factor.