  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.
