We show that it is possible to encode any communication protocol between two parties so that the protocol succeeds even if a (1/4 − ǫ) fraction of all symbols transmitted by the parties are corrupted adversarially, at a cost of increasing the communication in the protocol by a constant factor (the constant depends on epsilon). This encoding uses a constant sized alphabet. This improves on an earlier result of Schulman, who showed how to recover when the fraction of errors is bounded by 1/240.

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... also show how to simulate an arbitrary protocol with a protocol using the binary alphabet, a constant factor increase in communication and tolerating a 1/8 − ǫ fraction of errors. Introduction Suppose a sender wants to send an n bit message to a receiver, but some of the sender's transmissions may be received incorrectly. What is the best way to encode the message in order to recover from the errors? This question, first considered by Shannon [Sha48], initiated the study of error correcting codes, which have since found applications in many different contexts. The book [PWJ72] is a good reference. In our work, we study the analogous question in the more general setting of interactive communication. We refer the reader to the book [KN97] for an introduction to interactive communication protocols. Most models of computation, for example circuits, branching programs, streaming algorithms, distributed systems, inherently involve communication protocols, so the following question is well motivated -What is the best way to encode an arbitrary two party communication protocol, so that the protocol cannot be disrupted by errors in the communication? This more general problem was first considered by Schulman [Sch96]. There are several ways to model how errors may occur. Throughout this paper, we focus on the so called worst-case scenario; we only assume that the fraction of transmissions that are received incorrectly is bounded, and make no assumption about the distribution of errors 1 . For the case of one way communication considered by Shannon, it is known how to construct error correcting codes that can encode any n-bit message using O ǫ (n) bits, so that even if a (1/4 − ǫ) fraction of the transmitted bits are received incorrectly, the intended message can be recovered. If we allow the transmitted message to use symbols from an alphabet whose size depends on ǫ but remains independent of n, it is known how to encode the message so that it can be recovered even if a (1/2 − ǫ) fraction of all symbols are corrupted. Moreover, both the encoding and decoding can be implemented by efficient algorithms. It is impossible to recover from an error rate of 1/2 if one wants to recover the intended message exactly, since there can be no valid decoding of a received string that is equal to each of two valid messages half the time. In a general interactive protocol, each party must wait for a response from the other party before deciding on what to send in each round of communication, so it is not clear how to use error correcting codes to encode the communication of the protocol. If there are R rounds of communication, any encoding of the protocol that works by encoding each round of communication separately can be disrupted with an error rate of 1/2R -the errors can completely corrupt the shortest message in the protocol. Thus, this approach is not useful if R is * University of Toronto.