New pseudo-planar binomials in characteristic two and related schemes

TL;DR

This paper introduces three new classes of pseudo-planar binomials in characteristic two and explores their connection to association schemes on Galois rings, expanding the understanding of pseudo-planar functions.

Contribution

The paper presents novel classes of pseudo-planar binomials in characteristic two and demonstrates their link to association schemes on Galois rings, extending previous constructions.

Findings

Three new classes of pseudo-planar binomials in characteristic two are constructed.

Each pseudo-planar function induces an association scheme on a Galois ring.

The work broadens the scope of pseudo-planar functions and their applications.

Abstract

Planar functions in odd characteristic were introduced by Dembowski and Ostrom in order to construct finite projective planes in 1968. They were also used in the constructions of DES-like iterated ciphers, error-correcting codes, and signal sets. Recently, a new notion of pseudo-planar functions in even characteristic was proposed by Zhou. These new pseudo-planar functions, as an analogue of planar functions in odd characteristic, also bring about finite projective planes. There are three known infinite families of pseudo-planar monomial functions constructed by Schmidt and Zhou, and Scherr and Zieve. In this paper, three new classes of pseudo-planar binomials are provided. Moreover, we find that each pseudo-planar function gives an association scheme which is defined on a Galois ring.