Binary Error-Correcting Codes With Minimal Noiseless Feedback

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 $\approx \frac {1}{3}$ of Alice’s bits are flipped adversarially. This improves upon the classical setting without feedback, where recovery is not possible for error fractions exceeding $\frac {1}{4}$ . In the corresponding setting of erasures rather than bit flips, feedback improves the error resilience from $\frac {1}{2}-\epsilon $ to $1-\epsilon $ for any $\epsilon \gt 0$ . 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 $\epsilon \gt 0$ , we construct a coding scheme that tolerates a $ 1/3-\epsilon $ fraction of bit flips (respectively a $1-\epsilon $ fraction of erasures) 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 $1/4$ (respectively $1/2$ for erasures), and for schemes resilient to a $1/3-\epsilon $ fraction of bit flips (respectively a $1-\epsilon $ fraction of erasures), the number of rounds must grow as $\epsilon \to 0$ .