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/δ2+α) and rate I(W)−δ 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/δ2). This quadratic dependence on the gap δ to capacity is known to be the 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 binary erasure channel.