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In this paper, we investigate the problem of optimizing the expurgated upper bound to the probability of error associated with transmission over discrete memoryless channels. We find a general sufficient condition under which, for a given value of the parameter , the channel input distribution that leads to the optimal exponent corresponds to a constant memoryless source. We then derive a necessary and sufficient condition that the above property holds for all (even then, different values of o would, in general, induce different optimal input distributions). Finally, we define a class of equidistant channels that includes all binary input channels, and show that for this class and all the optimal expurgated exponent is attained by the uniform distribution over the inputs.