Systematic Transmission With Fountain Parity Checks for Erasure Channels With Stop Feedback

In this paper, we present new achievability bounds on the maximal achievable rate of variable-length stop-feedback (VLSF) codes operating over a binary erasure channel (BEC) at a fixed message size $M=2^k$. We provide new bounds for VLSF codes with zero error, infinite decoding times and with nonzero error, finite decoding times. Both new achievability bounds are proved by constructing a new VLSF code that employs systematic transmission of the first $k$ bits followed by random linear fountain parity bits decoded with a rank decoder. For VLSF codes with infinite decoding times, our new bound outperforms the state-of-the-art result for BEC by Devassy \emph{et al.} in 2016. We also give a negative answer to the open question Devassy \emph{et al.} put forward on whether the $23.4\mathrm{\%}$ backoff to capacity at $k=3$ is fundamental. For VLSF codes with finite decoding times, numerical evaluations show that the achievable rate for VLSF codes with a moderate number of decoding times closely approaches that for VLSF codes with infinite decoding times.