A lower bound on the error probability for signals in white Gaussian noise

In this correspondence we apply a recent inequality by de Caen (1997) to derive a lower bound on the probability of error for M-ary signals derived from a binary linear code and used on the additive white Gaussian noise channel with a maximum-likelihood decoder. This bound depends only on the weight enumerator of the code and the signal-to-noise ratio E_b/N₀. We show that this bound converges to the union upper bound as E_b/N₀ goes to infinity. Finally, by means of examples, we compare our lower bound with those of Shannon and Swaszek and with Poltyrev´s upper bound