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

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