A PTR polynomial for the Hughes planes and a new class of permutation polynomials involving Catalan numbers

TL;DR

This paper derives a polynomial representation for Hughes planes using PTR polynomials, revealing connections with Catalan numbers and introducing new classes of permutation polynomials with applications in cryptography.

Contribution

It provides the first polynomial representation of Hughes planes via PTR polynomials and introduces new permutation polynomial classes involving Catalan numbers.

Findings

Derived a reduced PTR polynomial for Hughes planes over nearfields.

Discovered new permutation polynomial classes involving Catalan numbers.

Analyzed the differential uniformity of the new permutation polynomials.

Abstract

Hughes introduced the projective planes that bear his name in 1957 and they have since been studied extensively. However, until now, no polynomial representation of a planar ternary ring that represents them has been determined. In this paper, we rectify this omission by determining a reduced PTR polynomial for any Hughes plane defined over a regular nearfield. The polynomials obtained provide a new surprising connection: both the Catalan numbers and generalized Catalan numbers occur among the coefficients, depending on the representation. Since every PTR polynomial has connections with several classes of permutation polynomials, we obtain three new infinite classes of permutation polynomials as a consequence of our main result, and these, too, involve the Catalan numbers. The differential uniformity of new permutation polynomials is also determined.