During this period, we studied two closely related topics: interactive communication with bounded memory and the construction of circuits resilient to noise.
We had significant progress on both of these topics: in the paper "Optimal Short-Circuit Resilient Formulas." We studied formulas(it is a weaker version of circuits) and figured out maximal noise resilience, showing matching lower and upper bounds. Next, in the paper "Interactive Coding with Small Memory" we have shown how to perform interactive communication when both parties have bounded memory. This result in turn leads us to the first quasi-polynomial construction of circuits resilient to the constant fraction of noise, which was one of the main goals of the research project.
During our research, we got a very surprising result which we did not expect to be true when we started the project: In the reliable transmission problem, a sender, Alice, wishes to transmit a bit-string x to a remote receiver, Bob, over a binary channel with adversarial noise. The solution to this problem is to encode x using an error correcting code. As it is long known that the distance of binary codes is at most 1/2, reliable transmission is possible only if the channel corrupts (flips) at most a 1/4-fraction of the communicated bits. We revisit the reliable transmission problem in the two-way setting, where both Alice and Bob can send bits to each other. Our main result is the construction of two-way error correcting codes that are resilient to a constant fraction of corruptions strictly larger than 1/4. Moreover, our code has constant rate and requires Bob to only send one short message
