Explicit constructions of Diophantine tuples over finite fields

A Diophantine $m$-tuple over a finite field ${\mathbb{F}}_q$ is a set $\{a_1,\dots ,a_m\}$ of $m$ distinct elements in ${\mathbb{F}}_q^{\ast }$ such that $a_i{a}_j+1$ is a square in ${\mathbb{F}}_q$ whenever $i\mathrm{\ne }j$. In this paper, we study $M(q)$, the maximum size of a Diophantine tuple over ${\mathbb{F}}_q$, assuming the characteristic of ${\mathbb{F}}_q$ is fixed and $q\to \mathrm{\infty }$. 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.