A Class of Transformations that Polarize Symmetric Binary-Input Memoryless Channels

TL;DR

This paper generalizes Arıkan's polar code construction by exploring transformations of the form G^{⊗ n} with an ℓ×ℓ matrix G, providing conditions for channel polarization and demonstrating that many such transformations achieve polarization.

Contribution

It introduces a broad class of transformations that ensure channel polarization, extending the applicability of polar codes beyond the original construction.

Findings

Large class of transformations polarize symmetric channels

Necessary and sufficient conditions for polarization

Extension of polar code construction methods

Abstract

A generalization of Ar\i kan's polar code construction using transformations of the form $G^{\otimes n}$ where $G$ is an $\ell \times \ell$ matrix is considered. Necessary and sufficient conditions are given for these transformations to ensure channel polarization. It is shown that a large class of such transformations polarize symmetric binary-input memoryless channels.