Explicit constructions of Diophantine tuples over finite fields

A Diophantine m -tuple over a finite field Fq F q is a set ₁,… , aₘ\ { a 1 , … , a m } of m distinct elements in Fq* F q ∗ such that aᵢaⱼ+1 a i a j + 1 is a square in Fq F q whenever i≠ j i ≠ j . In this paper, we study M ( q ), the maximum size of a Diophantine tuple over Fq F q , assuming the characteristic of Fq F q is fixed and q → ∞ 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.