On the degradations of Binary-Input Discrete Memoryless Channels

TL;DR

This paper investigates optimal degradation schemes for symmetric binary-input discrete memoryless channels to improve the evaluation of synthetic channels in polar code construction.

Contribution

It identifies all minimal-error degradations and proposes an algorithm to maximize symmetric capacity in channel degradation schemes.

Findings

All minimal-error degradations are characterized.

A necessary condition for maximum symmetric capacity degradations is established.

An efficient algorithm for finding optimal degradation schemes is proposed.

Abstract

For the polar codes introduced by Arikan in 2009, the first code family achieving the capacity of binary-input discrete memoryless channels (BIDMCs) with low-complexity encoding and decoding, it is crucial to evaluate the reliability of the synthetic channels resulted in the code construction. Since the synthetic channels have an output alphabet that grows exponentially with the code length, an effective method for faithfully evaluating their reliability is to replace them with degradations of manageable alphabet size. The main aim of this paper is to find the optimal degradations of symmetric BIDMCs. We determine all the degradations with the minimum probability of error decoding and give a necessary condition for the degradations with the maximum symmetric capacity. Finally, based on this necessary condition, we propose an efficient algorithm for finding degradation schemes that maximize the symmetric capacity.