A planar difference set is a difference set with parameter 
, so every nonzero element
of the ambient group occurs exactly once as a difference
of two elements of the set. If
the set has 
elements and the group
order is 
,
then its difference set order is 
and its parameters are
 |
In a cyclic group, a planar difference set can be represented by marks on a circle. For example, 
in 
divides a circle of circumference
7 into arcs of lengths 1, 2, and
4, as shown by the innermost ring above. Sums of one or two
consecutive arcs give every integer length from 1 through 6, while the full circumference
gives 7. The outer rings illustrate additional planar difference sets in cyclic
groups; their labels give the gaps between consecutive marks.
The 
translates of a planar difference set form a symmetric
block design with the same parameters and hence
give a finite projective plane of order
.
Singer (1938) proved that planar difference sets exist whenever 
is a prime power; the prime
power conjecture asserts that these are the only possible orders. In particular,
in 
is a planar difference set of order 2. A planar difference set in a cyclic
group is also called a perfect difference
set.
