# 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 • 10*^{6} for
abelian sets, *2 • 10*^{9} 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 • 10*^{10}. 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
- prime
- a product of twin primes
- 2
^{n }-1.

**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 H*^{T} = 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
*(4u*^{2}, 2u^{2} ± u, u^{2} ± u).
Schmidt showed the only
possible counterexamples with *k < 5 • 10*^{10}
are *u = 11715 or 82005*, and gave a heuristic argument that
counterexamples are unlikely.
### Barker Sequences

A Barker sequence is a sequence
{a_{1}, a_{2}, ... a_{v}} of
±1's for which ∑_{i=1}^{v-j}
|a_{i}a_{i+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 • 10*^{33}, 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 10^{100}.