Efficiency of Precode-Only Raptor codes and Turbo-Fountain Codes

This paper deals with precode-only (PCO) Raptor and Turbo fountain codes used on a noiseless channel, as it is the case for the BEC or the AWGN channel with high SNR. We find an expression for the performance of an ML-decoded PCO Raptor code. The performance parameter is the inefficiency of the code, i.e., the average amount of redundancy needed to decode the information bits. The inefficiency is expressed in terms of a generalized weight enumerating function. Also, we characterize the behavior of the Shannon ensemble in the same conditions. Finally, we extend the analysis to Turbo fountain codes, which are iteratively decoded. We obtain an upper bound to the Turbo code inefficiency when the deterministic interleaver is replaced by a uniform interleaver.