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 and their labels give the gaps between consecutive marks, as illustrated
in the figure above due to E. Pegg, Jr. (pers. comm., Aug. 18, 2026).
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.
