Explicit constructions of Diophantine tuples over finite fields

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