AWGN coding theorems from ensemble weight enumerators

We develop AWGN coding theorems for ensembles of codes for which we can calculate, or at least closely estimate, the ensemble weight enumerator. As a rule, for such an ensemble we can find a threshold c such that if E/sub b//N/sub 0/>c, then the ensemble maximum-likelihood error probability approaches zero. This threshold is always better, and usually much better, than can be obtained from the union bound. The role of low-weight code words is the key.