Spectra and Minimum Distances of Repeat Multiple-Accumulate Codes

In this paper, the ensembles of repeat multiple accumulate codes (RA(m)) which are obtained by interconnecting a repeater with a cascade of m accumulate codes through uniform random interleavers, are analyzed. It is proved that the average spectral shapes of these code ensembles are equal to 0 below a threshold distance epsilon(m) and, moreover, they form a nonincreasing sequence in m converging uniformly to the maximum between the average spectral shape of the linear random ensemble and Consequently the sequence epsilon(m) converges to the Gilbert-Varshamov (GV) distance. A further analysis allows to conclude that if m >= 2 the RA(m) are asymptotically good and that epsilon(m) is the typical normalized minimum distance when the interleaver length goes to infinity. Combining the two results it is possible to conclude that the typical distance of the ensembles RA(m) converges to the Gilbert-Varshamov bound.