Magnetoresistance of a two-dimensional electron gas in a random magnetic field

We report magnetoresistance measurements on a two-dimensional electron gas made from a high-mobility GaAs/${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As heterostructure, where the externally applied magnetic field was expelled from regions of the semiconductor by means of superconducting lead grains randomly distributed on the surface of the sample. A theoretical explanation in excellent agreement with the experiment is given within the framework of the semiclassical Boltzmann equation.