Spectra and minimum distances of repeat multiple accumulate codes

In this paper we consider ensembles of codes, denoted RAm, 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 isinm 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 isinm converges to the Gilbert-Varshamov distance. A further analysis allows to conclude that the threshold distance isinm is indeed the typical distance of the ensemble RAm when the interleaver length goes to infinity. Combining the two results we are able to conclude that the typical distance of the ensembles RAm converges to the Gilbert-Varshamov bound.

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