Planar functions, relative difference sets, and character theory

Abstract

Planar functions from ℤn to ℤn are studied in this paper. By investigating the character values of the corresponding relative difference sets, we obtain some nonexistence results of planar functions. In particular, we show that there is no planar functions from Zpq to ℤpq, where p and q are any primes. We also show that except for four undecided cases, there is no planar function from ℤn to ℤn if n is not a prime and n ≤ 50,000. © 1996 Academic Press, Inc.