The CompactLight design for a next-generation x-ray free-electron laser utilizes a C-band injector. This requires that the harmonic system used to linearize the beam's phase space must operate at X-band rf or higher. We investigate the optimum frequency for the harmonic system in the range of frequencies from 12 to 48 GHz. We describe the reasoning behind selecting 36 GHz (Ka-band) as our working harmonic frequency. The full linearizer system design including the power source, pulse compressor, and linearizing structure, along with options, is considered and presented. These designs are compared in terms of rf and beam dynamics performance. Two potential MW-level rf sources are discussed; a multibeam klystron and a gyro-klystron, while a klystron-based upconverter with an X-band driver is briefly discussed as an alternative path if even higher peak powers are needed. To further increase peak power, novel options for pulse compressors at Ka-band are discussed. Traveling and standing wave solutions for the structure are presented.