A family of Alltop functions that are EA-inequivalent to the cubic function

TL;DR

This paper introduces a new family of Alltop functions with applications in constructing sequence sets and mutually unbiased bases, expanding the known classes of such functions beyond the cubic case.

Contribution

The paper presents a novel family of Alltop functions that are EA-inequivalent to the cubic function, enhancing the toolkit for sequence and quantum basis constructions.

Findings

New family of Alltop functions identified

Alltop functions used to construct sequence sets

Alltop functions employed for mutually unbiased bases

Abstract

Sequences with optimal correlation properties are much sought after for applications in communication systems. In 1980, Alltop (IEEE Trans. Inf. Theory 26(3):350-354, 1980) described a set of sequences based on a cubic function and showed that these functions were optimal with respect to known bounds on auto and crosscorrelation. Subsequently these sequences were used to construct mutually unbiased bases, a structure of importance in quantum information theory. The key feature of this cubic function is that its difference function is a planar function. Functions with planar difference functions have been called \emph{Alltop functions}. This paper provides a new family of Alltop functions and establishes the use of Alltop functions for construction of sequence sets and MUBs.