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This paper shows that, under the average error probability formalism, the third-order term in the normal approximation for the additive white Gaussian noise channel with a maximal or equal power constraint is at least (1/2) log n + O(1). This improves on the lower bound by Polyanskiy-Poor-Verdú (2010) and matches the upper bound proved by the same authors.
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Tan et al. (2015) studied this question.
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