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

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.