Girth 5 Graphs from Elliptic Semiplanes

For 3 ≤ k ≤ 20 with k 6= 4, 8, 12, all the smallest currently known k–regular graphs of girth 5 have the same orders as the girth 5 graphs obtained by the following construction: take a (not necessarily Desarguesian) elliptic semiplane S of order n− 1 where n = k − r for some r ≥ 1; the Levi graph Γ (S) of S is an n–regular graph of girth 6; parallel classes of S induce co–cliques in Γ (S), some of which are eventually deleted; the remaining co–cliques are amalgamated with suitable r–regular graphs of girth at least 5. For k > 20, this construction yields some new instances underbidding the smallest orders known so far.