Spectra and minimum distances of repeat multiple accumulate codes

In this paper we consider ensembles of codes, denoted RA^m, obtained by a serial concatenation of a repetition code and m accumulate codes through uniform random inter-leavers. We analyze their average spectrum functions for each m showing that they are equal to 0 below a threshold distance isin_m and positive beyond it. One of our main results is to prove that these average spectrum functions form a not-increasing sequence in m converging uniformly to a limit spectrum function which is equal to the maximum between the average spectrum function of the classical linear random ensemble and 0. As a consequence the sequence isin_m converges to the Gilbert-Varshamov distance. A further analysis allows to conclude that the threshold distance isin_m is indeed the typical distance of the ensemble RA^m when the interleaver length goes to infinity. Combining the two results we are able to conclude that the typical distance of the ensembles RA^m converges to the Gilbert-Varshamov bound.