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In this paper we show how to attain the capacity of discrete symmetricchannels with polynomial time decoding complexity by considering iterated$(U|U+V)$ constructions with Reed-Solomon code or algebraic geometry codecomponents. These codes are decoded with a recursive computation of the {\em aposteriori} probabilities of the code symbols together with the Koetter-Vardysoft decoder used for decoding the code components in polynomial time. We showthat when the number of levels of the iterated $(U|U+V)$ construction tends toinfinity, we attain the capacity of any discrete symmetric channel in this way.This result follows from the polarization theorem together with a simple lemmaexplaining how the Koetter-Vardy decoder behaves for Reed-Solomon codes of rateclose to $1$. However, even if this way of attaining the capacity of asymmetric channel is essentially the Ar{\i}kan polarization theorem, there aresome differences with standard polar codes.  Indeed, with this strategy we can operate succesfully close to channelcapacity even with a small number of levels of the iterated $(U|U+V)$construction and the probability of error decays quasi-exponentially with thecodelength in such a case (i.e. exponentially if we forget about thelogarithmic terms in the exponent). We can even improve on this result byconsidering the algebraic geometry codes constructed in \cite{TVZ82}. In such acase, the probability of error decays exponentially in the codelength for anyrate below the capacity of the channel. Moreover, when comparing this strategyto Reed-Solomon codes (or more generally algebraic geometry codes) decoded withthe Koetter-Vardy decoding algorithm, it does not only improve the noise levelthat the code can tolerate, it also results in a significant complexity gain.

Attaining Capacity with Algebraic Geometry Codes through the $(U|U+V)$ Construction and Koetter-Vardy Soft Decoding