Properties of a certain stochastic dynamical system, channel polarization, and polar codes

Abstract

Abstract.A new family of codes, called polar codes, has recently been proposed by Arıkan. Polar codes are of theoretical importance because they are provably capacity achieving with low-complexity encoding and decoding. We first discuss basic properties of a certain stochastic dynamical system, on the basis of which properties of channel polarization and polar codes are reviewed, with emphasis on our recent results. IntroductionThe concept "More is different" is relevant not only in statistical physics but also in various other research fields, such as information theory, where one can find several examples in which properties of a system drastically change as the system size becomes large. As an example, channel coding theorem states that there exist error-correcting codes such that their error probabilities are arbitrarily small and yet their coding rates are arbitrarily close to the channel capacity. Although channel coding theorem is one of the most important theorems in information theory, it is also widely understood that channel coding theorem is a theorem of existence, so that one requires different arguments as to how one can construct practical codes with high performance. The problem of constructing codes which have practically low computational complexity in encoding and decoding, as well as good performance in view of channel coding theorem, remains a problem of particular importance in information theory and coding theory.Proposal of turbo codes in 1990s, as well as subsequent rediscovery of low-density parity-check (LDPC) codes, has ignited extensive research activities on the combination of codes with sparse graphical representation and probability-based iterative decoding algorithms. As a consequence, by now we have practical means of how to construct codes with low computational complexity and high performance. However, rigorous results justifying the high performance of LDPC codes are mostly limited to the case with erasure channels, and we have not arrived at a thorough understanding of LDPC codes for other channels.Arıkan [1] recently proposed polar codes as a class of error-correcting codes which is completely different from LDPC codes. Polar codes are theoretically very interesting because it saturates symmetric channel capacity (maximum mutual information when input distribution is assumed uniform, which is equal to the channel capacity for a symmetric channel) in the large codelength limit, whereas computational complexity of encoding and decoding is polynomial in the codelength. In this paper, we review recent developments regarding polar codes, as well as