On the asymptotic accuracy of the union bound

A new lower bound on the error probability of maximum likelihood decoding of a binary code on a binary symmetric channel was proved in Barg and McGregor (2004, cs.IT/0407011). It was observed in that paper that this bound leads to a new region of code rates in which the random coding exponent is asymptotically tight, giving a new region in which the reliability of the BSC is known exactly. The present paper explains the relation of these results to the union bound on the error probability.