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A simple derivation of the refined sphere packing bound under certain symmetry hypotheses

A judicious application of the Berry-Esseen theorem via suitable Augustin information measures is demonstrated to be sufficient for deriving the sphere packing bound with a prefactor that is Omega (n(-0.5(1-E'sp)((R)))) for all codes on certain families of channels (including the Gaussian channels and the nonstationary Renyi symmetric channels) and for the constant composition codes on stationary memoryless channels. The resulting nonasymptotic bounds have definite approximation error terms. As a preliminary result that might be of interest on its own, the trade-off between type I and type II error probabilities in the hypothesis testing problem with (possibly non-stationary) independent samples is determined up to some multiplicative constants, assuming that the probabilities of both types of error are decaying exponentially with the number of samples, using the Berry-Esseen theorem.