On the number of distinct values of a class of functions with finite domain

TL;DR

This paper establishes tight bounds on the number of distinct values a function with finite domain can have, with applications to finite fields, coding theory, and additive combinatorics, including a novel bound for planar functions.

Contribution

It introduces new bounds for the image set size of functions with finite domain, improving previous estimates and applying to finite field functions and related areas.

Findings

Derived bounds are best possible within elementary arguments.

Connected bounds to triangular numbers, suggesting potential improvements.

Provided the first non-trivial upper bound for planar functions over finite fields.

Abstract

By relating the number of images of a function with finite domain to a certain parameter, we obtain both an upper and lower bound for the image set. Even though the arguments are elementary, the bounds are, in some sense, best possible. The upper bound is also connected to triangular numbers, and a slight improvement to this bound could be obtained by resolving a problem on them. In the final section, we consider implications of our bounds in various settings, including finite fields, coding theory and additive combinatorics. In particular, we obtain the first non-trivial upper bound for the image set of a planar function over a finite field; this bound is better than the bound implied by the Dembowski-Ostrom conjecture.