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A 32-pole superconducting magnet with a 12 × 80 mm2 cold bore aperture was designed to serve as a multipole wiggler in the Taiwan synchrotron light source. The magnet consists of 32 pairs of racetrack NbTi superconducting coils with a periodic length of 60 mm, and can produce a maximum magnetic field of 3.2 Tesla at a pole gap of 18 mm. The superconducting coils, the aluminum-supporting block, and the return iron yokes are cooled to 4.4 K in LHe bath. The temperature of cold bore beam duct will be at 70 K using liquid nitrogen. Technical issues concerning the design of the magnet and its construction are discussed. A prototype magnet with five poles was also constructed to characterize the magnet design by means of various methods of magnetic field measurement.