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Planar functions, relative difference sets, and character theory. (English) Zbl 0865.05020
Let H and K be groups of order n. A mapping f from H to K is called a planar function of degree n if for each hH-{1}, the induced mapping f h :xf(hx)f(x) -1 is bijective. It is known that a planar function exists if and only if there exists an (n,n,n,1)-relative difference set in H×K relative to {1}×K. The author uses character theory to prove new results on the existence of planar functions from Z n to Z n and for the corresponding relative difference sets. In particular, the author shows that there are no planar functions from Z pq to Z pq where p and q are any primes and that except for 4 undecided cases, there is no planar function from Z n to Z n if n is not a prime and n50,000.
05B10Difference sets