On Bruen chains

It is known that a Bruen chain of the three-dimensional projective space PG(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⩽q⩽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)/2-cliques of a two related, yet distinct, undirected simple graphs.