A New Approach to Constructing Quadratic Pseudo-Planar Functions over $\gf_{2^n}$

TL;DR

This paper introduces a novel method for constructing quadratic pseudo-planar functions over finite fields of characteristic two, expanding the class of functions used in combinatorial designs and quantum information applications.

Contribution

It presents a new approach to constructing quadratic pseudo-planar functions and provides five explicit families, including binomials, trinomials, and quadrinomials, with generalizations of known functions.

Findings

Constructed five explicit families of pseudo-planar functions.

Revisited and generalized existing pseudo-planar functions.

Applications include projective planes, difference sets, presemifields, and optimal codebooks.

Abstract

Planar functions over finite fields give rise to finite projective planes. They were also used in the constructions of DES-like iterated ciphers, error-correcting codes, and codebooks. They were originally defined only in finite fields with odd characteristic, but recently Zhou introduced pesudo-planar functions in even characteristic which yields similar applications. All known pesudo-planar functions are quadratic and hence they give presemifields. In this paper, a new approach to constructing quadratic pseudo-planar functions is given. Then five explicit families of pseudo-planar functions are constructed, one of which is a binomial, two of which are trinomials, and the other two are quadrinomials. All known pesudo-planar functions are revisited, some of which are generalized. These functions not only lead to projective planes, relative difference sets and presemifields, but also give optimal codebooks meeting the Levenstein bound, complete sets of mutually unbiased bases (MUB) and compressed sensing matrices with low coherence.