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IEEE International Symposium on Information Theory (ISIT)

Attaining capacity with iterated (U|U + V) codes based on AG codes and Koetter-Vardy soft decoding

Irene Márquez-Corbella, Jean–Pierre Tillich

Published Jun 1, 2017• 6 citations
Publisher

Abstract

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 algebraic geometry (AG) code components. These codes are decoded with a recursive computation of the a posteriori probabilities of the code symbols together with decoding the AG components with the Koetter-Vardy algorithm. 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. Moreover the error probability decays quasi-exponentially with the codelength in the case of Reed-Solomon code constituents and exponentially with Tsfasman-Vladuts-Zink code constituents.

DOI
10.1109/isit.2017.8006479

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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 algebraic geometry (AG) code components. These codes are decoded with a recursive computation of the a posteriori probabilities of the code symbols together with decoding the AG components with the Koetter-Vardy algorithm. 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. Moreover the error probability decays quasi-exponentially with the codelength in the case of Reed-Solomon code constituents and exponentially with Tsfasman-Vladuts-Zink code constituents.
Attaining Capacity with Algebraic Geometry Codes through the (U∣U+V)(U|U+V)(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)(U|U+V)(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)(U|U+V)(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 111. However, even if this way of attaining the capacity of a symmetric channel is essentially the Arıkan 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)(U|U+V)(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.
Polar codes were introduced by Arikan in 2008 and are the first family of error-correcting codes achieving the symmetric capacity of an arbitrary binary-input discrete memoryless channel under low complexity encoding and using an efficient successive cancellation decoding strategy. Recently, non-binary polar codes have been studied, in which one can use different algebraic geometry codes to achieve better error decoding probability. In this paper, we study the performance of binary polar codes that are obtained from non-binary algebraic geometry codes using concatenation. For binary polar codes (i.e. binary kernels) of a given length nnn, we compare numerically the use of short algebraic geometry codes over large fields versus long algebraic geometry codes over small fields. We find that for each nnn there is an optimal choice. For binary kernels of size up to n≤1,800n \leq 1,800n≤1,800 a concatenated Reed-Solomon code outperforms other choices. For larger kernel sizes concatenated Hermitian codes or Suzuki codes will do better.
The decoding error probability of codes is studied as a function of their block length. It is shown that the existence of codes with a polynomially small decoding error probability implies the existence of codes with an exponentially small decoding error probability. Specifically, it is assumed that there exists a family of codes of length N and rate R=(1-ε)C (C is a capacity of a binary symmetric channel), whose decoding probability decreases polynomially in 1/N. It is shown that if the decoding probability decreases sufficiently fast, but still only polynomially fast in 1/N, then there exists another such family of codes whose decoding error probability decreases exponentially fast in N. Moreover, if the decoding time complexity of the assumed family of codes is polynomial in N and 1/ε, then the decoding time complexity of the presented family is linear in N and polynomial in 1/ε. These codes are compared to the recently presented codes of Barg and Zemor, ``Error Exponents of Expander Codes,'' IEEE Trans. Inform. Theory, 2002, and ``Concatenated Codes: Serial and Parallel,'' IEEE Trans. Inform. Theory, 2005. It is shown that the latter families can not be tuned to have exponentially decaying (in N) error probability, and at the same time to have decoding time complexity linear in N and polynomial in 1/ε.