Systems are provided for searching for a codeword from a plurality of codewords in a codebook for use in precoding, for example, as used in a multiple-input multiple-output (MIMO) transmission system. Dimension reduction techniques may be utilized to reduce the complexity and enhance the efficiency of the codebooks search. Null-spaces of an optimal codeword and codewords in a codebook may be computed. Distance values may be computed based on the null=spaces of the codewords. A codeword may be selected from the codebook based on a minimum distance value from the optimal codeword.