On Bruen chains

It is known that a Bruen chain of the three-dimensional projective space $\mathrm{P}\mathrm{G}(3,q)$ exists for every odd prime power $q$ at most $37$, except for $q=29$. It was shown by Cardinali et. al (2005) that Bruen chains do not exist for $41\le q\le 49$. We develop a model, based on finite fields, which allows us to extend this result to $41⩽q⩽97$, thereby adding more evidence to the conjecture that Bruen chains do not exist for $q>37$. Furthermore, we show that Bruen chains can be realised precisely as the $(q+1)\mathrm{/}2$-cliques of a two related, yet distinct, undirected simple graphs.