A generalized erasure channel in the sense of polarization for binary erasure channels

2018 IEEE International Symposium on Information Theory (ISIT)

Fujisaki

2018

Self Cite

This study examines multilevel channel polarization for a certain class of erasure channels that the input alphabet size is an arbitrary composite number. We derive asymptotic proportions of partially noiseless channels for such a class. The results of this study are proved by an argument of convergent sequences, inspired by Alsan and Telatar's simple proof of polarization [1], and without martingale convergence theorems for polarization process. I. INTRODUCTIONArıkan [2] proposed binary polar codes as a class of provable symmetric capacity achieving codes with deterministic constructions and low encoding/decoding complexity for binary-input discrete memoryless channels (DMCs). In non-binary polar codes, there are two types of channel polarization: strong polarization [3], [10] and multilevel polarization [4]-[8]. Strong polarization asymptotically makes similar extremal channels to binary cases, i.e., either noiseless or pure noisy. On the other hand, multilevel polarization allows to converse several types of partially noiseless channels. It was independently shown in [3]-[8], [10] that both strong and multilevel channel polarization can achieve the symmetric capacity by showing rate of polarization for the Bhattacharyya parameters. Although the asymptotic distributions of strong polarization are fully and simply characterized by the symmetric capacity, the asymptotic distribution of multilevel channel polarization is, however, still an open problem.Recently, the authors [9] proposed a certain class of erasure channels together with the recursive formulas of the polar transforms for such a class. In addition, we [9] also clarified the asymptotic distribution of multilevel channel polarization for such a class when the input alphabet size q is a prime power. In this paper, we examine further the asymptotic distribution for general composite numbers q.

Section: A Generalized Erasure Channelmentioning

confidence: 99%

Section: Multilevel Polarizationmentioning

confidence: 99%

“…The next subsection introduces such a class. 9 In [6,Theorem 6], the rate of polarization for Bhattacharyya parameter is also shown; but we omit it in the paper for simplicity.…”

Section: Multilevel Polarizationmentioning

confidence: 99%

Section: A Generalized Erasure Channelmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Asymptotic Distribution of Multilevel Channel Polarization for a Certain Class of Erasure Channels

2018 IEEE International Symposium on Information Theory (ISIT)

Fujisaki

2018

Self Cite

Countably Infinite Multilevel Source Polarization for Non-Stationary Erasure Distributions

2019 IEEE International Symposium on Information Theory (ISIT)

Fujisaki

2019

Self Cite

Polar transforms are central operations in the study of polar codes. This paper examines polar transforms for nonstationary memoryless sources on possibly infinite source alphabets. This is the first attempt of source polarization analysis over infinite alphabets. The source alphabet is defined to be a Polish group, and we handle the Arıkan-style two-by-two polar transform based on the group. Defining erasure distributions based on the normal subgroup structure, we give recursive formulas of the polar transform for our proposed erasure distributions. As a result, the recursive formulas lead to concrete examples of multilevel source polarization with countably infinite levels when the group is locally cyclic. We derive this result via elementary techniques in lattice theory.

A generalized erasure channel in the sense of polarization for binary erasure channels

2016 IEEE Information Theory Workshop (ITW)

2016

The polar transformation of a binary erasure channel (BEC) can be exactly approximated by other BECs. Arıkan proposed that polar codes for a BEC can be efficiently constructed by using its useful property. This study proposes a new class of arbitrary input generalized erasure channels, which can be exactly approximated the polar transformation by other same channel models, as with the BEC. One of the main results is the recursive formulas of the polar transformation of the proposed channel. In the study, we evaluate the polar transformation by using the α-mutual information. Particularly, when the input alphabet size is a prime power, we examines the following: (i) inequalities for the average of the α-mutual information of the proposed channel after the one-step polar transformation, and (ii) the exact proportion of polarizations of the α-mutual information of proposed channels in infinite number of polar transformations.