Coding Theorems for Repeat Multiple Accumulate Codes

In this paper the ensemble of codes formed by a serial concatenation of a repetition code with multiple accumulators connected through random interleavers is considered. 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 using the uniform interleaver approach. In accordance with earlier results in the literature, it is first shown that the minimum distance of repeat-accumulate codes can grow, at best, sublinearly with block length. Then, for repeat-accumulate-accumulate codes and rates of 1/3 or less, it is proved that these codes exhibit asymptotically linear distance growth with block length, where the gap to the Gilbert-Varshamov bound can be made vanishingly small by increasing the number of accumulators beyond two. In order to address larger rates, random puncturing of a low-rate mother code is introduced. It is shown that in this case the resulting ensemble of repeat-accumulate-accumulate codes asymptotically achieves linear distance growth close to the Gilbert-Varshamov bound. This holds even for very high rate codes.