The Rate of Interactive Codes Is Bounded Away from 1

Kol and Raz [STOC 2013] showed how to simulate any alternating two-party communication protocol designed to work over the noiseless channel, by a protocol that works over a stochastic channel that corrupts each sent symbol with probability epsilon > 0 independently, with only a 1 + O(root H(epsilon)) blowup to the communication. In particular, this implies that the maximum rate of such interactive codes approaches 1 as epsilon goes to 0, as is also the case for the maximum rate of classical error correcting codes. Over the past decade, followup works have strengthened and generalized this result to other noisy channels, stressing on how fast the rate approaches 1 as epsilon goes to 0, but retaining the assumption that the noiseless protocol is alternating. In this paper we consider the general case, where the noiseless protocols can have arbitrary orders of speaking. In contrast to Kol-Raz and to the followup results in this model, we show that the maximum rate of interactive codes that encode general protocols is upper bounded by a universal constant strictly smaller than 1. To put it differently, we show that there is an inherent blowup in communication when protocols with arbitrary orders of speaking are faced with any constant fraction of errors epsilon > 0. We mention that our result assumes a large alphabet set and resolves the (non-binary variant) of a conjecture by Haeupler [FOCS 2014].