Real-time oblivious erasure correction with linear time decoding and constant feedback

We continue the study of rateless codes for transmission of information across channels whose rate of erasure is unknown. In such a code, an infinite stream of encoding symbols can be generated from the message and sent across the erasure channel, and the decoder can decode the message after it has successfully collected a certain number of encoding symbols. A rateless erasure code is real-time oblivious if rather than collecting encoding symbols as they are received, the receiver either immediately decodes or discards each symbol it receives. Efficient real-time oblivious erasure correction uses a feedback channel in order to maximize the probability that a received encoding symbol is decoded rather than discarded. We construct codes which are real-time oblivious, but require fewer feedback messages and have faster decoding compared to previous work. Specifically, for a message of length $k'$, we improve the expected complexity of the feedback channel from $O(\sqrt{k'})$ to $O(1)$, and the expected decoding complexity from $O(k'\log(k'))$ to $O(k')$. Our method involves using an appropriate block erasure code to first encode the $k'$ message symbols, and then using a truncated version of the real-time oblivious erasure correction of Beimel et al (2007) to transmit the encoded message to the receiver, which then uses the decoding algorithm for the outer code to recover the message.