Capacity-Achieving Ensembles for the Binary Erasure Channel With Bounded Complexity

Abstract

We present two sequences of ensembles of non-systematic irregular repeat-accumulate codes which asymptotically (as their block length tends to infinity) achieve capacity on the binary erasure channel (BEC) with bounded complexity per information bit. This is in contrast to all previous constructions of capacity-achieving sequences of ensembles whose complexity grows at least like the log of the inverse of the gap (in rate) to capacity. The new bounded complexity result is achieved by puncturing bits, and allowing in this way a sufficient number of state nodes in the Tanner graph representing the codes. We also derive an informationtheoretic lower bound on the decoding complexity of randomly punctured codes on graphs. The bound holds for every memoryless binary-input output-symmetric channel and is refined for the BEC.

Error bounds for repeat-accumulate codes decoded via linear programming

“…In Table I we give the thresholds for the BSC transition probability for which the error bound (7) decays to zero, for some choices of q. It is apparent that the threshold worsens as q increases.…”

“…Let ǫ > 0 be some positive number. If the transition probability p satisfies p < q −4(ǫ+1+ 1 2 log q (4q−2)) , then when decoded using the RALP decoder, the code has word error probability WEP < K(log q n) · n −ǫ (7) where K is a positive constant.…”

“…Irregular RA code ensembles have been shown to achieve excellent performance under iterative message-passing decoding. For example, in the BEC there are known capacity-achieving sequences of codes (see e.g., [7]). Motivated by these results for iterative decoding, we examine regular and irregular RA codes under LP decoding.…”

On the growth rate of the weight distribution of irregular doubly-generalized LDPC codes

“…A local stopping set for this CN is a subset of the local code bits such that, if all of these bits are erased, MAP decoding cannot recover any of these bits. This occurs if and only if each column of G corresponding to erased bits is linearly independent of the columns of G corresponding to the non-erased bits 6 . The size of the local stopping set for the CN is equal to the number of erased local code bits.…”

Decoding complexity of irregular LDGM-LDPC codes over the BISOM channels

“…A lower-bound on the density of a parity-check matrix in terms of the multiplicativecapacity-gap ǫ was obtained in [3] and later tightened in [7]. For a code defined by a full-rank parity-check matrix, the lower-bound on the density is of [4] showed that non-systematic irregular-repeat-accumulate (NSIRA) codes could achieve rates arbitrarily close to the channel-capacity of a BISOM channel with bounded complexity. The rates close to channel-capacity were achieved by randomly puncturing the information bits of the NSIRA codes indepedently with a probability that depended on the gap to capacity.…”