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We consider random network coding in noisy large-scale multiple-access relay networks in which the source packets that are correctly received at a relay are linearly combined with randomly chosen coefficients and forwarded to the destination. We derive the union bound on the average probability of decoding error at the destination with the maximum likelihood decoding, averaged over all possible node locations and relay encoding rules. The union bound provides an upper bound to the probability of decoding error with the best network coding scheme and enables us to determine the error exponent. From the error exponent, we determine the reliability-rate tradeoff and the achievable rate at the high node density regime. The high node density analysis is useful for understanding the performance of large-scale multiple access relay networks. The energy saving at a source node offered by the energy expenditure at a relay node and the optimum constellation size that minimizes the energy per information bit (Eb/N0) are investigated as a function of reliability, rate, and node density. The effect of MIMO transmit modes at the relay nodes, when they are equipped with multiple antennas, on the network-wide reliability-rate tradeoff is investigated. The insight provided by the analysis is useful for understanding of the fundamental limit and tradeoffs in large-scale multiple-access relay networks with network coding.