Trapping Sets in Fountain Codes over Noisy Channels
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Fountain codes have demonstrated great results for the binary erasure channel and have already been incorporated into several international standards to recover lost packets at the application layer. These include multimedia broadcast/multicast sessions and digital video broadcasting on global internet-protocol. The rateless property of Fountain codes holds great promise for noisy channels. These are more sophisticated mathematical models representing errors on communications links rather than only erasures. The practical implementation of Fountain codes for these channels, however, is hampered by high decoding cost and delay. In this work we study trapping sets in Fountain codes over noisy channels and their effect on the decoding process. While trapping sets have received much attention for low-density parity-check (LDPC) codes, to our knowledge they have never been fully explored for Fountain codes. Our study takes into account the different code structure and the dynamic nature of Fountain codes. We show that 'error-free' trapping sets exist for Fountain codes. When the decoder is caught in an error-free trapping set it actually has the correct message estimate, but is unable to detect this is the case. Thus, the decoding process continues, increasing the decoding cost and delay for naught. The decoding process for rateless codes consists of one or more decoding attempts. We show that trapping sets may reappear as part of other trapping sets on subsequent decoding attempts or be defeated by the reception of more symbols. Based on our observations we propose early termination methods that use trapping set detection to obtain improvements in realized rate, latency, and decoding cost for Fountain codes.