Arikan 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 α > 0. We construct, for any sufficiently small δ > 0, binary linear codes of block length O(1/δ 2+α ) and rate I(W )δ that enable reliable communication on W with quasilinear 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/δ 2 ). This quadratic dependence on the gap δ 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.