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Codebook based limited feedback is a practical way to obtain partial channel state information at the transmitter in a precoded multiple-input multiple-output wireless system. In this correspondence, we propose a systematic codebook design. The proposed Kerdock codebook consists of multiple mutually unbiased unitary basis matrices with quaternary alphabet and the identity matrix. We propose to derive the beamforming and precoding codebooks from this base codebook, eliminating the requirement to store multiple codebooks. Furthermore, we show that quaternary alphabet facilitates efficient codebook storage and codeword search. We derive the chordal distance for two and four antenna codebooks, showing that the proposed codebooks compare favorably with prior designs. Monte Carlo simulations compare achievable rates and error rates for different codebooks sizes.