Parity-check density versus performance of binary linear block codes over memoryless symmetric channels

Abstract

Low-density parity-check (LDPC) codes are efficiently encoded and decoded due to the sparseness of their parity-check matrices. Motivated by their remarkable performance and feasible complexity under iterative message-passing decoding, we derive lower bounds on the density of parity-check matrices of binary linear codes whose transmission takes place over a memoryless binary-input output-symmetric (MBIOS) channel. The bounds are expressed in terms of the gap between the rate of these codes for which reliable communications is achievable and the channel capacity; they are valid for every sequence of binary linear block codes. For every MBIOS channel, we construct a sequence of ensembles of regular LDPC codes, so that an upper bound on the asymptotic density of their parity-check matrices scales similarly to the lower bound. The tightness of the lower bound is demonstrated for the binary erasure channel by analyzing a sequence of ensembles of right-regular LDPC codes which was introduced by Shokrollahi, and which is known to achieve the capacity of this channel. Under iterative message-passing decoding, we show that this sequence of ensembles is asymptotically optimal (in a sense to be defined in this paper), strengthening a result of Shokrollahi. Finally, we derive lower bounds on the bit error probability and on the gap to capacity for binary linear block codes which are represented by bipartite graphs, and study their performance limitations over MBIOS channels. The latter bounds provide a quantitative measure for the number of cycles of bipartite graphs which represent good error-correction codes.

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]).…”

“…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.…”

On Achievable Rates and Complexity of LDPC Codes Over Parallel Channels: Bounds and Applications

“…The latter result is used for the derivation of lower bounds on the decoding complexity of randomly and intentionally punctured LDPC codes for MBIOS channels; looser versions of these bounds suggest a simplified re-derivation of previously reported bounds on the decoding complexity of randomly punctured LDPC codes (as shown in the appendix). Finally, Section VI summarizes our discussion, and presents a diagram which shows interconnections between the theorems introduced in this paper and some other previously reported results from [1], [9], [10], [12], [14], [17]. The preliminary material on ensembles of LDPC codes and notation required for this paper are introduced in [13] and [17,Section 2].…”

“…where the first transition is based on (14) and follows along the same lines as [17, Appendix B.2]), and the second transition is due to the fact that for all i ∈ I(j), the pdf of the random variable Ω i is independent of i, see (4). Summing over all the parity-check equations of H gives…”