Sending a Message with Unknown Noise

TL;DR

This paper presents an algorithm for reliable message transmission over a two-way channel with unknown and adversarial bit-flip noise, achieving near-optimal efficiency in terms of bits sent.

Contribution

The authors introduce a new method for communicating reliably with unknown noise levels and message length, optimizing the total bits sent.

Findings

Expected bits sent is close to message length plus noise-dependent overhead.

Achieves asymptotic optimality when noise is large or small.

Handles unknown message length and noise level efficiently.

Abstract

Alice and Bob are connected via a two-way channel, and Alice wants to send a message of $L$ bits to Bob. An adversary flips an arbitrary but finite number of bits, $T$, on the channel. This adversary knows our algorithm and Alice's message, but does not know any private random bits generated by Alice or Bob, nor the bits sent over the channel, except when these bits can be predicted by knowledge of Alice's message or our algorithm. We want Bob to receive Alice's message and for both players to terminate, with error probability at most $\delta > 0$, where $\delta$ is a parameter known to both Alice and Bob. Unfortunately, the value $T$ is unknown in advance to either Alice or Bob, and the value $L$ is unknown in advance to Bob. We describe an algorithm to solve the above problem while sending an expected $L + O \left( T + \min \left(T+1,\frac{L}{\log L} \right) \log \left( \frac{L}{\delta} \right) \right)$ bits. A special case is when $\delta = O(1/L^c)$, for some constant $c$. Then when $T = o(L/\log L)$, the expected number of bits sent is $L + o(L)$, and when $T = \Omega(L)$, the expected number of bits sent is $L + O\left( T \right)$, which is asymptotically optimal.