Existence of further equivalent starter-block orders

Determine whether any pairs of multiplicative-subgroup orders beyond (5, 10) and (13, 26) have the same set of prime parameters for which the corresponding pairs (q, k) and (q, k') yield 3-designs under the action of PSL(2, q).

Background

The paper studies when the orbit of the multiplicative subgroup of order k in F_q, viewed as a subset of PG(1,q), forms the block set of a 3-design under PSL(2,q). It proves two equivalences: for q ≡ 1 (mod 20), the orders 5 and 10 give 3-designs simultaneously, and for q ≡ 1 (mod 52), the same holds for the orders 13 and 26.

Computational searches identified no additional pairs of subgroup orders with this matching behavior. Thus, beyond the two families established in the paper, it remains unresolved whether other pairs of orders exhibit an analogous equivalence.

References

The result mentioned in the previous paragraph is also formulated as Theorem 4.1: (q, 5) gives a 3-design if and only if (q, 10) gives a 3-design for q ≡ 1 (mod 20). In Theorem 5.1, we show that (q, 13) gives a 3-design if and only if (q, 26) gives a 3-design for q ≡ 1 (mod 52). By computer search, we could not find any more such pair of k.

— 3-Designs from PSL(2,q) with cyclic starter blocks  (2502.13331 - Hanaki et al., 18 Feb 2025) in Introduction, p. 2; discussion following Example 3.2, p. 6