Compositional inverses and complete mappings over finite fields

We study compositional inverses of permutation polynomials and complete mappings over finite fields. Recently the compositional inverses of linearized permutation binomials were obtained in Wu (2013). In this paper we obtain compositional inverses of a class of linearized binomials permuting the kernel of the trace map. It was also shown in Tuxanidy and Wang (2014) that computing inverses of bijections of subspaces has an application in determining the compositional inverses of certain permutation classes related to linearized polynomials. Consequently, we give the compositional inverse of a new class of complete mappings. This complete mapping class extends several recent constructions given in Laigle-Chapuy (2007), Samardjiska and Gligoroski (2014), Wu and Lin (2013), Wu and Lin (2015), Wu et al. (2013). We also construct recursively a class of complete mappings involving multi-trace functions.