General Strong Polarization

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