Explicit constructions of Diophantine tuples over finite fields

A Diophantine m-tuple over a finite field \({\mathbb F}_q\) is a set \(\{a_1,\ldots , a_m\}\) of m distinct elements in \(\mathbb {F}_{q}^{*}\) such that \(a_{i}a_{j}+1\) is a square in \({\mathbb F}_q\) whenever \(i\ne j\). 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 \(q \rightarrow \infty \). 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.