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Zbl 0865.05020
Ma, S.L.
Planar functions, relative difference sets, and character theory.
(English)
[J] J. Algebra 185, No.2, 342-356 (1996). ISSN 0021-8693

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 $h\in H-\{1\}$, the induced mapping $f_h : x \rightarrow f(hx)f(x)^{-1}$ 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 $H\times K$ relative to $\{1\} \times K$. 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 $n\leq 50,000$.
MSC 2000:
*05B10 Difference sets

Keywords: planar function; difference set; character theory

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