Accumulate–Repeat–Accumulate Codes: Capacity-Achieving Ensembles of Systematic Codes for the Erasure Channel With Bounded Complexity

Abstract

The paper introduces ensembles of accumulate-repeat-accumulate (ARA) codes which asymptotically achieve capacity on the binary erasure channel (BEC) with bounded complexity, per information bit, of encoding and decoding. It also introduces symmetry properties which play a central role in the construction of capacity-achieving ensembles for the BEC with bounded complexity. The results here improve on the tradeoff between performance and complexity provided by previous constructions of capacity-achieving ensembles of codes defined on graphs. The superiority of ARA codes with moderate to large block length is exemplified by computer simulations which compare their performance with those of previously reported capacity-achieving ensembles of LDPC and IRA codes. The ARA codes also have the advantage of being systematic.

Bounds on the number of iterations for turbo-like code ensembles over the binary erasure channel

“…For various families of code ensembles, Table I compares the number of iterations and the graphical complexity which are required to achieve a given fraction 1 − ε (where ε can be made arbitrarily small) of the capacity of a BEC with vanishing bit erasure probability. The results in Table I are based on lower bounds and some achievability results which are related to the graphical complexity of various families of code ensembles defined on graphs (see [8], [9], [11], [12]); the results related to the number of iterations are based on the lower bounds introduced here (for rigorous proofs, see [13] families of code ensembles (including LDPC codes, systematic and non-systematic IRA codes, and ARA codes), the number of iterations which are required to achieve a fixed bit erasure probability scales at least like the inverse of the gap between the channel capacity and the design rate of the ensemble. This conclusion holds provided that the fraction of degree-2 variable nodes in the Tanner graph does not vanish as the gap to capacity vanishes (where under mild conditions, this property is satisfied for sequences of capacity-achieving LDPC code ensembles, see [10,Lemma 5]).…”

“…Ensembles of irregular and systematic ARA codes, which asymptotically achieve the capacity of the BEC with bounded graphical complexity, are presented in [9]. This bounded complexity result stays in contrast to LDPC code ensembles, which have been shown to require unbounded graphical complexity in order to approach channel capacity, even under maximumlikelihood decoding (see [11]).…”

“…This bounded complexity result stays in contrast to LDPC code ensembles, which have been shown to require unbounded graphical complexity in order to approach channel capacity, even under maximumlikelihood decoding (see [11]). The setting and notation used to characterize ensembles of irregular and systematic ARA codes and their iterative decoders are introduced in [9, Section II] (due to space limitations, we refer the reader to [9]). …”

“…In the asymptotic case where the block length tends to infinity and the transmission takes place over the BEC, suitable constructions of capacity-achieving systematic ARA ensembles enable a fundamentally improved tradeoff between their graphical complexity and their achievable gap (in rate) to capacity under iterative decoding, as compared to LDPC code ensembles (see [9], [11]). This raises the question whether the number of iterations required to achieve a desired bit erasure probability under iterative decoding, can be reduced by using systematic ARA ensembles.…”

Upper bounds on the MAP threshold of iterative decoding systems with erasure noise

“…ARA codes were introduced by Abbasfar, Divsalar, and Kung in [7]. Later, it was shown that the DE analysis of IRA and ARA codes can be reduced to the DE analysis of LDPC codes via a technique known as graph reduction [8].…”