In this paper we consider the ensemble of codes formed by a serial concatenation of a repetition code with mul-tiple accumulators through uniform random interleavers. Based on finite length weight enumerators for these codes, asymptotic expressions for the minimum distance and an arbitrary number of accumulators larger than one are derived. In accordance with earlier results in the literature, we irst show that the minimum distance of RA codes can grow, at best, sublinearly with the block length. Then, for RAA codes and rates of 1/3 or smaller, it is proved that these codes exhibit linear distance growth with block length. where the gap to the Gilbert-Varshamov bound can be made arbitrarily small by increasing the number of accumulators beyond two. In order to address rates larger than 1/3, random puncturing of a low-rate mother code is introduced. We show that in this case the resulting ensemble of EAA codes asymptotically achieves linear distance growth close to the Gilbert-Varshamov bound. This holds even for very high rate codes.