General Strong Polarization

TL;DR

This paper extends the theory of polarization to all matrices satisfying a necessary condition, providing simpler proofs and enabling efficient capacity-achieving codes for all symmetric memoryless channels with small error probabilities.

Contribution

It generalizes strong polarization results to all matrices meeting the necessary condition, simplifying proofs and broadening applicability to arbitrary symmetric channels.

Findings

Strong polarization holds over all prime fields.

Efficient codes achieve exponentially small error probabilities.

Lengths are inverse polynomial in the gap to capacity.

Abstract

Arikan's exciting discovery of polar codes has provided an altogether new way to efficiently achieve Shannon capacity. Given a (constant-sized) invertible matrix $M$, a family of polar codes can be associated with this matrix and its ability to approach capacity follows from the {\em polarization} of an associated $[0,1]$-bounded martingale, namely its convergence in the limit to either $0$ or $1$. Arikan showed polarization of the martingale associated with the matrix $G_2 = \left(\begin{matrix} 1& 0 1& 1\end{matrix}\right)$ to get capacity achieving codes. His analysis was later extended to all matrices $M$ that satisfy an obvious necessary condition for polarization. While Arikan's theorem does not guarantee that the codes achieve capacity at small blocklengths, it turns out that a "strong" analysis of the polarization of the underlying martingale would lead to such constructions. Indeed for the martingale associated with $G_2$ such a strong polarization was shown in two independent works ([Guruswami and Xia, IEEE IT '15] and [Hassani et al., IEEE IT '14]), resolving a major theoretical challenge of the efficient attainment of Shannon capacity. In this work we extend the result above to cover martingales associated with all matrices that satisfy the necessary condition for (weak) polarization. In addition to being vastly more general, our proofs of strong polarization are also simpler and modular. Specifically, our result shows strong polarization over all prime fields and leads to efficient capacity-achieving codes for arbitrary symmetric memoryless channels. We show how to use our analyses to achieve exponentially small error probabilities at lengths inverse polynomial in the gap to capacity. Indeed we show that we can essentially match any error probability with lengths that are only inverse polynomial in the gap to capacity.