Exact evaluation of the Random coding Error probability and Error Exponent

The Clipped union bound is generally used when coding theorems are proved with random coding. As noted by Shulman and Feder [1, Lemma A.2], if the codewords are triple-wise independent, the bound is tight up to factor 12 . In this paper we calculate the exact bound as a function of a CDF of the pairwise error of any word. The method assumes a general metric-based decoder and general channel. Evaluation of the Error Exponent is demonstrated for the mismatched decoding case.