Intermediate Performance of Rateless Codes

Fountain codes are designed so that all input symbols can be recovered from a slightly larger number of coded symbols, with high probability using an iterative decoder. In this paper we investigate the number of input symbols that can be recovered by the same decoder, but when the number of coded symbols available is less than the total number of input symbols. Of course recovery of all inputs is not possible, and the fraction that can be recovered will depend on the output degree distribution of the code. In this paper we (a) outer bound the fraction of inputs that can be recovered for any output degree distribution of the code, and (b) design degree distributions which meet/perform close to this bound. Our results are of interest for real-time systems using rateless codes, and for Raptor-type two-stage designs.