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As the integration level of superconducting digital circuits increases, flux trapping in these devices becomes a serious problem. High resolution A/D converters and other high speed signal processing systems have been demonstrated with junction counts well into the 10/sup 3/ range. Such large circuits require special testing techniques to prevent flux trapping within the gates, which can reduce bias margins and cause malfunctions of these devices. We discuss the results of experiments using single flux quantum shift registers in which we have varied the ground plane hole pattern and magnetic shield degaussing procedure to minimize flux trapping in these circuits. The operating bias margins of the shift registers have been measured as a function of different testing procedures and ground plane hole designs. In situ degaussing of the magnetic shields aids in the reduction of flux trapping and gave the best results. Measurements of the permeability of mu metal at 4.2 K are discussed.