Open Questions

There are a large number of open conjectures about difference sets, with various levels of evidence for. Here are some of the major ones. All are stated for for (v,k,λ) difference sets D in a group G with order n=k-λ .

Prime Power Conjecture

If D is a planar abelian difference set (i.e. λ = 1), then n is a prime power.

Evidence for:True up to 2 • 106 for abelian sets, 2 • 109 for cyclic.

Ryser's Conjecture

If G is cyclic, gcd(v,n) = 1.

Evidence for:True for all k ≤ 107, and for all parameters of known difference sets with k ≤ 5 • 1010. Turyn showed that if a self-conjugate prime p divides gcd(v,n), then there is no cyclic (v,k,λ) difference set. This would imply no circulant Hadamard matrices or Barker sequences.

Lander's Conjecture

If G is abelian, and p divides gcd(v,n), then the Sylow p-subgroup of G cannot be cyclic.

Evidence for:Leung, Ma and Schmidt proved the conjecture for n a power of a prime > 3.

McFarland's Conjecture

The only difference sets with multiplier -1 are (4000,775,150) and circulant Hadamard matrices.

Hall's Multiplier Conjecture

Any prime p dividing n and not v is a multiplier of D.

Evidence for:The First Multiplier Theorem shows this is true for p > λ. Other multiplier theorems deal with some cases.

Cyclic Hadamard Difference Sets

Golomb and Song conjectured that all cyclic difference sets with parameters (4n-1,2n-1,n-1) (called cyclic Hadamard difference sets) have v either
Evidence for: All but seven possible counterexamples with v ≤ 10000 have been eliminated.

Circulant Hadamard Matrices

A circulant Hadamard matrix is an n x n matrix H of ± 1's with cyclic symmetry for which H HT = nI. It is conjectured that they exist only for n=1,4. Such a matrix is equivalent to a difference set, which must have parameters (4u2, 2u2 ± u, u2 ± u). Schmidt showed the only possible counterexamples with k < 5 • 1010 are u = 11715 or 82005, and gave a heuristic argument that counterexamples are unlikely.

Barker Sequences

A Barker sequence is a sequence {a1, a2, ... av} of ±1's for which ∑i=1v-j |aiai+j| ≤ 1 for j=1,...,v-1.

Evidence for: Extensive computations by Leung and Schmidt, and recently Borwein and Mossinghoff showed that there are none of length 13 < l < 4 • 1033, with the possible exception of 3979201339721749133016171583224100. A 2015 preprint of Leung and Schmidt eliminated this exception, as well as 229,682 of the 237,807 open cases less than 10100.