Sending a Message with Unknown Noise
This solves a communication problem with adversarial noise for distributed systems, but it is incremental as it builds on known error-correcting and coding techniques.
The paper tackles the problem of sending a message over a noisy channel with an unknown number of bit flips, achieving an algorithm with an expected bit cost of L + O(T + min(T+1, L/log L) log(L/δ)), which is asymptotically optimal in certain cases.
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 $δ> 0$, where $δ$ 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 $L + O \left( T + \min \left(T+1,\frac{L}{\log L} \right) \log \left( \frac{L}δ \right) \right)$ bits. A special case is when $δ= O(1/L^c)$, for some constant $c$. Then when $T = o(L/\log L)$, the expected number of bits sent is $L + o(L)$, and when $T = Ω(L)$, the expected number of bits sent is $L + O\left( T \right)$, which is asymptotically optimal.