Abstract

Polyanskiy Abstract—It is demonstrated that codewords of good codes for the additive white Gaussian noise (AWGN) channel become more and more isotropically distributed (in the sense of evalu-ating quadratic forms) and resemble white Gaussian noise (in the sense of ℓp norms) as the code approaches closer to the fundamental limits. In particular, it is shown that the optimal Gaussian code must necessarily have peak-to-average power ratio (PAPR) of order log n. Index Terms—Shannon theory, empirical statistics, channel capacity, channel dispersion, concentration of measure, additive white Gaussian noise. I.