From what I've read I've gathered the following facts:

• There are seven known such graphs.
• Certain parameter sets are ruled out by the Krein conditions and the absolute bound.
• Beyond that, little or nothing is known.

Am I missing something? I have read Biggs's report which lists all small feasible parameter sets and apparently this paper shows that $(324,57,0,12)$ is infeasible.

Is something else known about this problem?