$\beta$-expansion: A Theoretical Framework for Fast and Recursive Construction of Polar Codes

TL;DR

This paper introduces $eta$-expansion as a novel theoretical framework for the fast and recursive construction of polar codes, leveraging polynomial equations and universal partial order to improve efficiency.

Contribution

It proposes a new $eta$-expansion method for polar code construction, enabling recursive, low-complexity code design with proven asymptotic properties.

Findings

Interval for $eta$ converges to approximately 1.1892 for large block lengths

The $eta$-expansion preserves nested frozen set properties

Simulation results validate the theoretical framework

Abstract

In this work, we introduce $\beta$-expansion, a notion borrowed from number theory, as a theoretical framework to study fast construction of polar codes based on a recursive structure of universal partial order (UPO) and polarization weight (PW) algorithm. We show that polar codes can be recursively constructed from UPO by continuously solving several polynomial equations at each recursive step. From these polynomial equations, we can extract an interval for $\beta$, such that ranking the synthetic channels through a closed-form $\beta$-expansion preserves the property of nested frozen sets, which is a desired feature for low-complex construction. In an example of AWGN channels, we show that this interval for $\beta$ converges to a constant close to $1.1892 \approx 2^{1/4}$ when the code block-length trends to infinity. Both asymptotic analysis and simulation results validate our theoretical claims.