Difference Set

Let $G$ be a Group of Order $h$ and $D$ be a set of $k$ elements of $G$. If the set of differences $d_i-d_j$ contains every Nonzero element of $G$ exactly $\lambda$ times, then $D$ is a $(h,k,\lambda)$-difference set in $G$ of Order $n=k-\lambda$. If $\lambda=1$, the difference set is called planar. The quadratic residues in the Galois Field $GF(11)$ form a difference set. If there is a difference set of size $k$ in a group $G$, then $2{k\choose 2}$ must be a multiple of $\vert G\vert-1$, where ${k\choose 2}$ is a Binomial Coefficient.

See also Bruck-Ryser-Chowla Theorem, First Multiplier Theorem, Prime Power Conjecture

References

Gordon, D. M. ``The Prime Power Conjecture is True for $n<2,000,000$.'' Electronic J. Combinatorics 1, R6 1-7, 1994. http://www.combinatorics.org/Volume_1/volume1.html#R6.