Potent Tree Codes and their applications: Coding for Interactive Communication, revisited

In this work, we study the fundamental problem of reliable interactive communication over a noisy channel. In a breakthrough sequence of papers published in 1992 and 1993 [Sch92, Sch93], Schulman gave non-constructive proofs of the existence of general methods to emulate any two-party interactive protocol such that: (1) the emulation protocol only takes a constantfactor longer than the original protocol, and (2) if the emulation protocol is executed over a noisy channel (BSC), then the probability that the emulation protocol fails to perfectly emulate the original protocol is exponentially small in the total length of the protocol. Unfortunately, Schulman’s emulation procedures either only work in a model with a large amount of shared randomness [Sch92], or are non-constructive in that they rely on the existence of good tree codes [Sch93]. The only known proofs of the existence of good tree codes are non-constructive, and finding an explicit construction remains an important open problem. Indeed, randomly generated tree codes are not good tree codes with overwhelming probability. In this work, we revisit the problem of reliable interactive communication, and obtain the following results: • We introduce a new notion of goodness for a tree code, and define the notion of a potent tree code. We believe that this notion is of independent interest. • We prove the correctness of an explicit emulation procedure based on any potent tree code. (This replaces the need for good tree codes in the work of Schulman [Sch93].) • We show that a randomly generated tree code (with suitable constant alphabet size) is a potent tree code with overwhelming probability. Furthermore we are able to partially derandomize this result using only O(n) random bits, where n is the depth of the tree. These (derandomized) results allow us to obtain the first fully explicit emulation procedure for reliable interactive communication over noisy channels with a constant communication overhead, with failure probability that is exponentially small in the length of the original communication protocol. Our results also extend to the case of interactive multi-party communication among a constant number of parties.