Determine whether the Bodmann–Haas minimum is nearly attained by known Sidon sets

Determine whether the quantity m(d), defined as the smallest order of a finite abelian group containing a Sidon set of size d, is nearly attained by subsets of the Erdős–Turán, Singer, Bose, Spence, or Hughes Sidon sets listed in Proposition 9, in order to assess whether the Bodmann–Haas construction can yield substantially better bounds on n(d).

Background

The paper defines m(d) as the size of the smallest finite abelian group containing a Sidon set of size d. Corollary 8 bounds the minimum size n(d) of a weighted projective 2-design by n(d) ≤ m(d)+d, using the Bodmann–Haas construction applied to Sidon sets.

The authors state that their construction is limited if m(d) is already nearly achieved by subsets of the known Sidon-set families in Proposition 9. Resolving whether this near-attainment holds would clarify whether improvements to the design bound can come from better analysis of the existing construction or require a fundamentally different construction.