Planar Functions, Relative Difference Sets, and Character Theory Original Research Article

Planar functions from2nto3nare 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 fromZpqto4pq, wherepandqare any primes. We also show that except for four undecided cases, there is no planar function from5nto6nifnis not a prime andn≤50,000.