On the Arikan Transformations of Binary-Input Discrete Memoryless Channels

TL;DR

This paper develops an algebraic method to analyze synthetic channels in polar codes, enabling better evaluation of their reliability and providing bounds on their likelihood ratios.

Contribution

It introduces a novel algebraic approach to characterize synthetic channels in polar codes and derives bounds on their likelihood ratio groups.

Findings

Synthetic channels can be characterized as random switching channels of BSCs.

A lower bound on the average number of likelihood ratio groups is established.

The method improves the evaluation of synthetic channel reliability.

Abstract

The polar codes introduced by Arikan in 2009 achieve the capacity of binary-input discrete memoryless channels (BIDMCs) with low complexity encoding and decoding. Identifying the unreliable synthetic channels, generated by Arikan transformation during the construction of these polar codes, is crucial. Currently, because of the large size of the output alphabets of synthetic channels, there is no efficient and practical approach to evaluate their reliability in general. To tackle this problem, by converting the generation of synthetic channels in polar code construction into algebraic operations, in this paper we develop a method to characterize the synthetic channels as random switching channels of binary symmetric channels when the underlying channels are symmetric. Moreover, a lower bound for the average number of elements that possess the same likelihood ratio within the output alphabet of any synthetic channel generated in polar codes is also derived.