On the Synthetic Channels in Polar Codes over Binary-Input Discrete Memoryless Channels

TL;DR

This paper introduces a novel algebraic approach to evaluate the reliability of synthetic channels in polar codes over binary-input discrete memoryless channels, addressing the challenge of large output alphabets.

Contribution

It defines equivalence and symmetry for synthetic channels, characterizes symmetric channels as RSCs, and derives compact representations and bounds for likelihood ratio elements.

Findings

Characterization of symmetric BIDMCs as RSCs

Development of algebraic operations for synthetic channel generation

Lower bound on likelihood ratio element count in synthetic channels

Abstract

Polar codes introduced by Arikan in 2009 are the first code family achieving the capacity of binary-input discrete memoryless channels (BIDMCs) with low-complexity encoding and decoding. Identifying unreliable synthetic channels in polar code construction is crucial. Currently, because of the large size of the output alphabets of synthetic channels, there is no effective approach to evaluate their reliability, except in the case that the underlying channels are binary erasure channels. This paper defines equivalence and symmetry based on the likelihood ratio profile of BIDMCs and characterizes symmetric BIDMCs as random switching channels (RSCs) of binary symmetric channels. By converting the generation of synthetic channels in polar code construction into algebraic operations on underlying channels, some compact representations of RSCs for these synthetic channels are derived. 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.