We consider the task of multiparty computation performed over networks in the presence of random noise. Given an n-party protocol that takes R rounds assuming noiseless communication, the goal is to find a coding scheme that takes R rounds and computes the same function with high probability even when the communication is noisy, while maintaining a constant asymptotical rate, i.e., while keeping lim n,R→∞ R/R positive. Rajagopalan and Schulman (STOC '94) were the first to consider this

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... and provided a coding scheme with rate O(1/ log(d + 1)), where d is the maximal degree of connectivity in the network. While that scheme provides a constant rate coding for many practical situations, in the worst case, e.g., when the network is a complete graph, the rate is O(1/ log n), which tends to 0 as n tends to infinity. We revisit this question and provide an efficient coding scheme with a constant rate for the interesting case of fully connected networks. We furthermore extend the result and show that if the network has mixing time m, then there exists an efficient coding scheme with rate O(1/m 3 log m). This implies a constant rate coding scheme for any n-party protocol over a network with a constant mixing time, and in particular for random graphs with n vertices and degrees n Ω(1) .