A binary erasure channel with erasure probability which can take one of two values is considered. Transmission is done by using a low density parity-check code under the requirement that completely successful decoding is possible when the channel is in its better state, while tolerating some predetermined residual erasure fraction when the channel is in its worse state. Upper bounds on the achievable design rate under iterative decoding are derived for this setting. These bounds are compared to rates obtained by practical code profiles. It is also observed that when exceeding the capacity of the erasure channel, the performance of such codes exhibits graceful degradation as measured by the residual erasure fraction.

A binary erasure channel with erasure probability which can take one of two values is considered. Transmission is done by using a low density parity-check code under the requirement that completely successful decoding is possible when the channel is in its better state, while tolerating some predetermined residual erasure fraction when the channel is in its worse state. Upper bounds on the achievable design rate under iterative decoding are derived for this setting. These bounds are compared to rates obtained by practical code profiles. It is also observed that when exceeding the capacity of the erasure channel, the performance of such codes exhibits graceful degradation as measured by the residual erasure fraction.

The ensemble of regular low-definition parity-check (LDPC) codes is considered. Using concentration results on the weight distribution, lower bounds on the error rate of a random code in the ensemble are derived. These bounds hold with some confidence level. Combining these results with known lower bounds on the error exponent, confidence intervals on the error exponent, under maximum-likelihood (ML) decoding, are obtained. Over a large range of channel parameter and transmission rate values, when the graph connectivity is sufficiently large, the upper bound of the interval approaches the lower bound, and the probability that the error exponent is within the interval can be arbitrarily close to one. In fact, in this case the true error exponent approaches the maximum between the random coding and the expurgated random coding exponents, with probability that approaches one.

We consider the ensemble of regular LDPC codes and use recent concentration results on the distance spectrum to derive upper bounds on the error exponent of a randomly chosen code from the ensemble. These bounds hold with some confidence level that approaches one as the connectivity of the graph increases. We show that the bounds can be used to obtain the true error exponent over some range of channel parameter values, with the above confidence level

We consider the ensemble of regular LDPC codes and obtain an expression for the second moment of the distance spectrum. We show how this expression can be used to derive a lower bound on the probability that the growth rate of a randomly chosen code from the ensemble is equal to the growth rate of the average distance spectrum, when the block length is sufficiently large. In particular, when the connectivity of the code is sufficiently large, the distance spectrum of a code in the ensemble is concentrated. We then derive a lower bound on the probability (confidence level) that the minimum distance and error rate, respectively, of a randomly chosen code from the ensemble are upper and lower bounded by some values (which depend on the confidence level)

We derive upper bounds on the maximum achievable rate of low-density parity-check (LDPC) codes used over the binary erasure channel (BEC) under Gallager's decoding algorithm, given their right-degree distribution. We demonstrate the bounds on the ensemble of right-regular LDPC codes and compare them with an explicit left-degree distribution constructed from the given right degree.