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

TL;DR

This paper derives new bounds on the maximum achievable rate of variable-length stop-feedback codes over erasure channels, introducing a systematic transmission scheme with fountain parity checks that improves existing results and addresses open questions.

Contribution

It introduces a new VLSF coding scheme with fountain parity checks, providing improved achievability bounds and resolving an open question about capacity backoff.

Findings

New bounds outperform previous results for infinite decoding times.

The proposed scheme closely approaches infinite decoding time performance with moderate decoding times.

The open question on the 23.4% backoff at k=3 is answered negatively.

Abstract

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 $23.4\%$ 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.