  We study compositional inverses of permutation polynomials, complete
mappings, mutually orthogonal Latin squares, and bent vectorial functions.
Recently it was obtained in [33] the compositional inverses of linearized
permutation binomials over finite fields. It was also noted in [29] that
computing inverses of bijections of subspaces have applications in determining
the compositional inverses of certain permutation classes related to linearized
polynomials. In this paper we obtain compositional inverses of a class of
linearized binomials permuting the kernel of the trace map. As an application
of this result, we give the compositional inverse of a class of complete
mappings. This complete mapping class improves upon a recent construction given
in [34]. We also construct recursively a class of complete mappings involving
multi-trace functions. Finally we use these complete mappings to derive a set
of mutually orthogonal Latin squares, and to construct a class of $p$-ary bent
vectorial functions from the Maiorana-McFarland class.
