Probability of correct decoding for a channel with additive normal noise and a binary symmetric channel with a random choice of code words

We consider the problem of estimating the probabilities of right decoding of messages consisting of zeros and ones in the case where zero is coded by a vector randomly chosen with equal probabilities from the set of all binary vectors of length $m$ of even weight and the one is coded by such a vector of length $m$ of odd weight. It is shown that the results for the model with additive Gaussian noise can significantly differ from the results for the simplified binary model.