McFarland’s classification conjecture for reversible abelian difference sets

Classify all reversible abelian difference sets by proving or disproving McFarland’s conjecture that every reversible abelian difference set has parameters of the form (4u², 2u² ± u, u² ± u) for some integer u ≥ 1, or has parameters (4000, 775, 150) or (4000, 3225, 2600).

Background

The paper relates λ-fold near-factorizations with |G| = s + t to difference sets: a difference set with parameters (v, k, λ) yields a near-factorization with parameters (k, v − k) and multiplicity k − λ. Symmetric near-factorizations correspond specifically to reversible difference sets, namely difference sets invariant under taking inverses.

The authors state McFarland’s conjecture in the context of reversible abelian difference sets. Resolving this conjecture would determine which parameter sets can arise from reversible abelian difference sets and thereby clarify a substantial family of symmetric λ-fold near-factorizations.