This paper investigates polar coding schemes achieving capacity for the AWGN channel. The approaches using a multiple access channel with a large number of binary-input users and a single-user channel with a large prime-cardinality input are compared with respect to complexity attributes. The problem of finding discrete approximations to the Gaussian input is then investigated, and it is shown that a quantile quantizer achieves a gap to capacity which decreases like 1/q (where q is the number of constellation points), improving on the 1/log(q) decay achieved with a binomial (central limit theorem) quantizer.
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