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We report values of the giant magnetoresistance (GMR) effect exceeding 21% in symmetric spin valves, the highest values ever reported for such structures. The key elements in this achievement are the use of a Co/Cu/Co/Cu/Co multilayer in which the center Co layer is substantially thicker than the outer Co layers and the use of the antiferromagnetic insulator NiO at the top and bottom to pin the adjacent Co layers magnetically. The relative Co layer thicknesses suggest that some specular scattering of conduction electrons may occur at the metal/insulator interfaces and may enhance the GMR.