In order to achieve rate-compatible RM codes, two most effective ways are puncturing and shortening. In this paper, puncturing schemes are studied. Different from the existing spherical puncturing, puncturing schemes which do no change the Plotkin structure of RM codes are proposed, enabling existing decoding techniques be readily accessible. The proposed puncturing schemes are based on Hamming weights of column indices of the generator matrix of RM codes. When Hamming weights of column indices (also determining the column weights, CW) are the same, different strategies are proposed, producing four different puncturing strategies, called CW-IV, CW-LSB, CW-MSB, and CW-MSB-Sym, respectively. Analysis is performed to show the union bound on error performance of RM codes with puncturing. Theoretically, it also shows that the punctured first 1st-order subcode can have potentially better performance than the non-puncturing case if puncturing is properly designed. Simulation results show that the proposed puncturing strategies outperform random and quasi-uniform puncturing (QUP) schemes in terms of block error rate (BLER). Union bound results also confirm that the first 1st-order subcode shows better performance than the original non-puncturing case. This fact indicates a more efficient transmission scheme of RM codes: transmitting part of the original codeword to increase the spectrum efficiency while achieving a better BLER performance under recursive list decoding.