Nonasymptotic coding-rate bounds for binary erasure channels with feedback

TL;DR

This paper derives nonasymptotic bounds on the maximum coding rate for binary erasure channels with feedback, using novel coding schemes and hypothesis testing, highlighting significant gaps in certain scenarios.

Contribution

It introduces new achievability and converse bounds for variable-length feedback codes over BECs, including a scheme based on Huffman coding and an analysis of fountain codes.

Findings

Achievability and converse bounds match at zero error probability.

Significant gap (23%) in small message regimes with feedback.

Bounds depend on channel erasure probability and message size.

Abstract

We present nonasymptotic achievability and converse bounds on the maximum coding rate (for a fixed average error probability and a fixed average blocklength) of variable-length full-feedback (VLF) and variable-length stop-feedback (VLSF) codes operating over a binary erasure channel (BEC). For the VLF setup, the achievability bound relies on a scheme that maps each message onto a variable-length Huffman codeword and then repeats each bit of the codeword until it is received correctly. The converse bound is inspired by the meta-converse framework by Polyanskiy, Poor, and Verd\'u (2010) and relies on binary sequential hypothesis testing. For the case of zero error probability, our achievability and converse bounds match. For the VLSF case, we provide achievability bounds that exploit the following feature of BEC: the decoder can assess the correctness of its estimate by verifying whether the chosen codeword is the only one that is compatible with the erasure pattern. One of these bounds is obtained by analyzing the performance of a variable-length extension of random linear fountain codes. The gap between the VLSF achievability and the VLF converse bound, when number of messages is small, is significant: $23\%$ for 8 messages on a BEC with erasure probability $0.5.$ The absence of a tight VLSF converse bound does not allow us to assess whether this gap is fundamental.