Planar Difference Functions

TL;DR

This paper investigates the existence of Alltop type functions in finite fields, proving their non-existence in characteristic 3 and identifying $x^3$ as the unique known planar function of this type.

Contribution

It establishes non-existence results for Alltop type functions in characteristic 3 and characterizes the only known planar function of this type within a specific class.

Findings

Alltop type functions do not exist in characteristic 3 fields.

The only known planar function of this type in a certain class is $x^3$.

Provides theoretical constraints on the construction of sequences with specific correlation properties.

Abstract

In 1980 Alltop produced a family of cubic phase sequences that nearly meet the Welch bound for maximum non-peak correlation magnitude. This family of sequences were shown by Wooters and Fields to be useful for quantum state tomography. Alltop's construction used a function that is not planar, but whose difference function is planar. In this paper we show that Alltop type functions cannot exist in fields of characteristic 3 and that for a known class of planar functions, $x^3$ is the only Alltop type function.