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 question, 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 (1) .