Perfect Difference Set

A Set of Residues $\{a_1, a_2, \ldots, a_{k+1}\}$ (mod $n$) such that every Nonzero Residue can be uniquely expressed in the form $a_i-a_j$. Examples include $\{1, 2, 4\}$ (mod 7) and $\{1, 2, 5, 7\}$ (mod 13). A Necessary condition for a difference set to exist is that $n$ be of the form $k^2+k+1$. A Sufficient condition is that $k$ be a Prime Power. Perfect sets can be used in the construction of Perfect Rulers.

See also Perfect Ruler

References

Guy, R. K. ``Modular Difference Sets and Error Correcting Codes.'' §C10 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 118-121, 1994.