In this article a primal­dual interior­point method for solving linear programs is proposed. A new approach for generating higher­order search directions, and a new method for an efficient higher­order subspace search along several search directions are the basis of the proposed extension. The subspace search is reduced to a linear program in several variables. The method using the simplest (two­dimensional) subprograms is a slight variation of Mehrotra's predictor­corrector method, and is thus known to be practically very efficient. The higher­dimensional subproblems are guaranteed to be at least as effective --- with respect to a certain measure --- as the two­dimensional ones. Numerical experiments with the PCx­package indicate that also in practice the higher order subspace search is very cheap and efficient.