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A key component of multiuser MIMO using zero-forcing precoding is the feedback of quantized channel state information to the base station. A problem arises when each user has a common codebook as the quantized channels can form a singular matrix that results in a reduced sum-rate. In this paper, we propose two new structured constructions to generate different codebooks at each user via transformations of a base codebook. The first construction is based on the Householder transform, which is used to construct a different codebook at each user for most types of base codebooks, with no storage in addition to the base codebook. A feature of our first construction is that the transformed codebook using the Fourier base codebook has a search complexity reduction of up to 50% compared to the standard approach, although only one additional unique codebook can be constructed with this type of base codebook. To construct multiple different codebooks using the Fourier base codebook, we propose a second construction that is based on the representation theory of groups. We show that both our constructions significantly reduce storage requirements compared with the intuitive but impractical random construction, while obtaining the same sum-rate performance. In particular, we only require elements generated directly from the base codebook or from finite fields, instead of random complex numbers.