Topologically simple, totally disconnected, locally compact infinite matrix groups

We construct uncountably many non-locally isomorphic examples of topologically simple nondiscrete totally disconnected locally compact groups. The new examples differ from known examples of such groups in that they have trivial quasi-centre, but also have infinite abelian locally normal subgroups. The examples are constructed as almost upper-triangular matrices modulo scalar matrices over finite fields, where 'almost upper-triangular' is defined with respect to one of an uncountable family of preorders generalising the orders (Z, ≤) and (N, ≤).