Binary Error-Correcting Codes with Minimal Noiseless Feedback

TL;DR

This paper develops and analyzes error-correcting codes with limited feedback, achieving near-optimal error tolerance with minimal feedback bits and rounds, and establishes matching lower bounds for feedback requirements.

Contribution

It introduces coding schemes that tolerate high error rates with logarithmic feedback bits and proves lower bounds on feedback needed for certain error thresholds.

Findings

Achieves error correction for up to 1/3 - epsilon fraction of errors with O(log k) feedback bits.

Proves that Omega(log k) feedback bits are necessary for error correction above 1/4 error fraction.

Shows that for erasures, O(log k) feedback bits suffice to tolerate nearly all erasures, with bounds matching the error case.

Abstract

In the setting of error-correcting codes with feedback, Alice wishes to communicate a $k$-bit message $x$ to Bob by sending a sequence of bits over a channel while noiselessly receiving feedback from Bob. It has been long known (Berlekamp, 1964) that in this model, Bob can still correctly determine $x$ even if $\approx \frac13$ of Alice's bits are flipped adversarially. This improves upon the classical setting without feedback, where recovery is not possible for error fractions exceeding $\frac14$. The original feedback setting assumes that after transmitting each bit, Alice knows (via feedback) what bit Bob received. In this work, our focus in on the limited feedback model, where Bob is only allowed to send a few bits at a small number of pre-designated points in the protocol. For any desired $\epsilon > 0$, we construct a coding scheme that tolerates a fraction $ 1/3-\epsilon$ of bit flips relying only on $O_\epsilon(\log k)$ bits of feedback from Bob sent in a fixed $O_\epsilon(1)$ number of rounds. We complement this with a matching lower bound showing that $\Omega(\log k)$ bits of feedback are necessary to recover from an error fraction exceeding $1/4$ (the threshold without any feedback), and for schemes resilient to a fraction $1/3-\epsilon$ of bit flips, the number of rounds must grow as $\epsilon \to 0$. We also study (and resolve) the question for the simpler model of erasures. We show that $O_\epsilon(\log k)$ bits of feedback spread over $O_\epsilon(1)$ rounds suffice to tolerate a fraction $(1-\epsilon)$ of erasures. Likewise, our $\Omega(\log k)$ lower bound applies for erasure fractions exceeding $1/2$, and an increasing number of rounds are required as the erasure fraction approaches $1$.