On the Construction of Polar Codes

TL;DR

This paper presents an efficient framework for constructing polar codes over BMS channels, enabling nearly linear complexity algorithms to identify good channels with high accuracy, improving the practicality of polar code design.

Contribution

The paper introduces a unified framework to analyze and improve the complexity and accuracy of approximate polar code construction algorithms.

Findings

Algorithms can find almost all good channels with near-linear complexity.

Numerical and analytical results demonstrate high efficiency of the proposed methods.

Construction complexity is nearly linear in block length, with only a polylogarithmic factor.

Abstract

We consider the problem of efficiently constructing polar codes over binary memoryless symmetric (BMS) channels. The complexity of designing polar codes via an exact evaluation of the polarized channels to find which ones are "good" appears to be exponential in the block length. In \cite{TV11}, Tal and Vardy show that if instead the evaluation if performed approximately, the construction has only linear complexity. In this paper, we follow this approach and present a framework where the algorithms of \cite{TV11} and new related algorithms can be analyzed for complexity and accuracy. We provide numerical and analytical results on the efficiency of such algorithms, in particular we show that one can find all the "good" channels (except a vanishing fraction) with almost linear complexity in block-length (except a polylogarithmic factor).