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Consider a problem of forward error-correction for the Additive White Gaussian Noise (AWGN) channel. For finite block-length codes the back-off from the channel capacity is inversely proportional to the square root of the block-length. In this paper it is shown that codes achieving this tradeoff must necessarily have Peak-to-Average Power Ratio (PAPR) proportional to logarithm of the block-length. This is extended to codes approaching capacity slower, and to PAPR measured at the output of an OFDM modulator. As a by-product the convergence of (Smith's) amplitude-constrained AWGN capacity to Shannon's classical formula is characterized in the regime of large amplitudes. This converse-type result builds upon recent contributions in the paper of empirical output distributions of good channel codes.
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