We investigate the Rubinstein-Duke model for polymer reptation by means of density-matrix renormalization group techniques both in the absence and presence of a driving field. In the former case the renewal time tau and the diffusion coefficient D are calculated for chains up to N=150 reptons and their scaling behavior in N is analyzed. Both quantities scale as powers of N:tausimilar toN(z) and Dsimilar to1/N-x with the asymptotic exponents z=3 and x=2, in agreement with the reptation theory. For an intermediate range of lengths, however, the data are well fitted by some effective exponents whose values are quite sensitive to the dynamics of the end reptons. We find 2.7