Joint error correction enhancement of the fountain codes concept

Fountain codes like LT or Raptor codes, also known as rateless erasure codes, allow to encode a message as some number of packets, such that any large enough subset of these packets is sufficient to fully reconstruct the message. It requires undamaged packets, while the packets which were not lost are usually damaged in real scenarios. Hence, an additional error correction layer is often required: adding some level of redundancy to each packet to be able to repair eventual damages. This approach requires a priori knowledge of the final damage level of every packet - insufficient redundancy leads to packet loss, overprotection means suboptimal channel rate. However, the sender may have inaccurate or even no a priori information about the final damage levels, for example in applications like broadcasting, degradation of a storage medium or damage of picture watermarking. Joint Reconstruction Codes (JRC) setting is introduced and discussed in this paper for the purpose of removing the need of a priori knowledge of damage level and sub-optimality caused by overprotection and discarding underprotected packets. It is obtained by combining both processes: reconstruction from multiple packets and forward error correction. The decoder combines the resultant informational content of all received packets accordingly to their actual noise level, which can be estimated a posteriori individually for each packet. Assuming binary symmetric channel (BSC) of $\epsilon$ bit-flip probability, every potentially damaged bit carries $R_0(\epsilon)=1-h_1(\epsilon)$ bits of information, where $h_1$ is the Shannon entropy. The minimal requirement to fully reconstruct the message is that the sum of rate $R_0(\epsilon)$ over all bits is at least the size of the message. We will discuss sequential decoding for the reconstruction purpose, which statistical behavior can be estimated using Renyi entropy.