Fountain Codes under Maximum Likelihood Decoding

This dissertation focuses on fountain codes under maximum likelihood (ML) decoding. Fountain codes are a class of erasure correcting codes that can generate an endless amount of coded symbols and were conceived to deliver data files over data networks to a potentially large population of users. First Luby transform (LT) codes are considered, which represent the first class of practical fountain codes. Concretely, the focus is on LT codes under inactivation decoding, an efficient ML decoding algorithm that is widely used in practical systems. More precisely, the decoding complexity of LT codes under inactivation decoding is analyzed in terms of the expected number of inactivations. The proposed analysis is based on a dynamical programming approach. This analysis is then extended to provide the probability distribution of the number of inactivations. Additionally a lower complexity approximate analysis is introduced and a code design example is presented that illustrates how these analysis techniques can be used to design LT codes. Next Raptor codes under ML decoding are considered. An upper bound to the probability of decoding failure of q-ary Raptor codes is developed, considering the weight enumerator of the outer code (precode). The bound is shown to be tight, specially in the error floor region, by means of simulations. This bound shows how Raptor codes can be analyzed similarly to a traditional serial concatenation of (fixed-rate) block codes. Next, a heuristic method is presented that yields an approximate analysis of Raptor codes under inactivation decoding. It is also shown by means of an example how the results in this thesis can be used to design Raptor codes. Raptor codes are next analyzed in a fixed-rate setting. Concretely, a Raptor code ensemble with an outer code picked from the linear random ensemble is considered. For this ensemble, the average weight enumerator and its growth rate are provided. Furthermore, sufficient and necessary conditions for the ensemble to have a minimum distance growing linearly with the block length are presented. The ensemble analyzed resembles standard Raptor codes, as it is shown by means of simulations. Finally a new class of fountain codes is introduced, that consists of a parallel concatenation of a block code with a linear random fountain code (LRFC). This scheme is specially interesting when the block code is a maximum distance separable (MDS) code. In this case, the scheme can provide failure probabilities lower than those of LRFC codes by several orders of magnitude, provided that the erasure probability of the channel is not too high.