Arıkan Meets Shannon: Polar Codes With Near-Optimal Convergence to Channel Capacity

Let
$W$
be a binary-input memoryless symmetric (BMS) channel with Shannon capacity
$I(W)$
and fix any
$\alpha > 0$
. We construct, for any sufficiently small
$\delta > 0$
, binary linear codes of block length
$O(1/\delta ^{2+\alpha })$
and rate
$I(W)-\delta $
that enable reliable communication on
$W$
with quasi-linear time encoding and decoding. Shannon’s noisy coding theorem established the
existence
of such codes (without efficient constructions or decoding) with block length
$O(1/\delta ^{2})$
. This quadratic dependence on the gap
$\delta $
to capacity is known to be best possible. Our result thus yields a constructive version of Shannon’s theorem with near-optimal convergence to capacity as a function of the block length. This resolves a central theoretical challenge associated with the attainment of Shannon capacity. Previously such a result was only known for the erasure channel. Our codes are a variant of Arıkan’s polar codes based on multiple carefully constructed local kernels, one for each intermediate channel that arises in the decoding. A crucial ingredient in the analysis is a strong converse of the noisy coding theorem when communicating using random linear codes on arbitrary BMS channels. Our converse theorem shows extreme unpredictability of even a single message bit for random coding at rates slightly above capacity.