Let f : GF(p)n → GF(p). When p = 2, Bernasconi and Codenotti discovered a correspondence between certain properties of f (e.g., if it is bent) and properties of its associated Cayley graph. Analogously, but much earlier, Dillon showed that f is bent if and only if the “level curves” of f have certain combinatorial properties (again, only when p = 2). We investigate an analogous theory when p > 2. We formulate some problems concerning natural generalizations of the Bernasconi correspondence and Dillon correspondence. We give a partial classification, in a combinatorial way, of even bent functions f : GF(p)n → GF(p) with f(0) = 0 for (p, n) = (3, 2), (3, 3), and (5, 2), where “even” means f(x) = f(?x). We will show that for any prime p > 2, there are (p+ 1)!/2 amorphic bent functions f : GF(p)2 → GF(p) of signature (p ? 1, p ? 1,...,p ? 1) with algebraic normal form that is homogeneous of degree p ? 1. They are all weakly regular. (Briefly, an amorphic bent function is one whose edge-weighted Cayley graph corresponds to an amorphic association scheme.) Our main conjecture is Conjecture 2, but a number of other open questions are scattered throughout the paper.
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