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