More on the number of distinct values of a class of functions

TL;DR

This paper investigates the limits of the number of distinct values for certain functions over finite fields, showing that known bounds are not tight for planar functions and providing new bounds through optimization and number theory techniques.

Contribution

The authors prove that the upper bounds for the number of distinct values cannot be tight for planar functions and larger classes, and they develop an algorithmic approach to determine tighter bounds.

Findings

Upper bounds are not tight for planar functions over finite fields.

A complete solution to a partitioning optimization problem is provided.

New upper bounds are derived using Diophantine equations and class number theory.

Abstract

In a previous article the authors determined the best-known upper bound for the cardinality of the image set for several classes of functions, including planar functions. Here, we show that the upper bound cannot be tight for planar functions over finite fields. This follows from a more general result proving that the upper bound cannot be tight for a much larger class of functions over an abelian group of order $y^n$ with $n>1$. Moreover, the tightness of the upper bound for the larger class of functions is equivalent to the existence of planar difference sets. To obtain better upper bounds, we first completely resolve an optimization problem involving the partitioning of a number into triangular parts. Our solution, which is algorithmic and constructive, allows us to determine tight upper bounds provided the relevant parameters are given explicitly. We also provide a suite of upper bounds which can be applied across a range of parameters. These are established via a well-studied Diophantine equation and are related to class numbers of quadratic number fields.