Uniqueness of the PSL(2,13) group association scheme

Determine whether the group association scheme of PSL(2,13) is uniquely determined up to combinatorial isomorphism by its intersection numbers, and establish whether a different switching construction can produce a counterexample.

Background

The switching construction for PSL(2,q) produces counterexamples for many prime powers but does not produce a new partition for q=13. The cases q=5 and q=7 are known to be uniquely determined, while PSL(2,9) is covered by the paper’s A_6 construction.

Consequently, among the excluded non-abelian simple cases considered in this discussion, q=13 is the remaining unresolved case. The authors specifically leave open both the uniqueness question and the possibility that another switching construction could yield a counterexample.

References

Thus only $q=13$ remains open. Is the group association scheme $\mathfrak X(PSL(2,13))$ uniquely determined up to combinatorial isomorphism by its intersection numbers? In particular, can a different switching construction yield a counterexample?

— Twisted primitive group association schemes  (2608.16278 - Higashitani et al., 17 Aug 2026) in Section 6, “Summary and future questions”