Explorations of edge-weighted Cayley graphs and p-ary bent functions

TL;DR

This paper explores the relationship between p-ary bent functions and their associated Cayley graphs, extending known binary results to larger primes using weighted partial difference sets and computational classification.

Contribution

It introduces a new approach using weighted partial difference sets to analyze p-ary bent functions and investigates their properties through computational classification in small cases.

Findings

Identifies limitations of direct generalizations from p=2 to p>2.

Proposes a conjecture relating function properties to combinatorial structures.

Provides classifications for small prime cases through extensive computations.

Abstract

Let f be a function mapping an n dimensional vector space over GF(p) to GF(p). When p is 2, Bernasconi et al. have shown that there is 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 had certain combinatorial properties (again, only when p is 2). The attempt is to investigate an analogous theory when p is greater than 2 using the (apparently new) combinatorial concept of a weighted partial difference set. More precisely, we try to investigate which properties of the Cayley graph of f can be characterized in terms of function-theoretic properties of f, and which function-theoretic properties of f correspond to combinatorial properties of the set of "level curves", i.e., the inverse map of f. While the natural generalizations of the Bernasconi correspondence and Dillon correspondence are not true in general, using extensive computations, we are able to determine a classification in some small cases. Our main conjecture is Conjecture 67.