Difference Set - Basic Facts

Basic Facts

• A simple counting argument shows that there are exactly pairs of elements from that will yield nonidentity elements, so every difference set must satisfy the equation .
• If is a difference set, and, then is also a difference set, and is called a translate of .
• The set of all translates of a difference set forms a symmetric block design. In such a design there are elements (mostly called points) and blocks. Each block of the design consists of points, each point is contained in blocks. Any two blocks have exactly elements in common and any two points are "joined" by blocks. The group then acts as an automorphism group of the design. It is sharply transitive on points and blocks.
  • In particular, if, then the difference set gives rise to a projective plane. An example of a (7,3,1) difference set in the group is the subset . The translates of this difference set gives the Fano plane.
• Since every difference set gives a symmetric design, the parameter set must satisfy the Bruck–Chowla–Ryser theorem.
• Not every symmetric design gives a difference set.