Arikan meets Shannon: polar codes with near-optimal convergence to channel capacity

Abstract

Let be a binary-input memoryless symmetric (BMS) channel with Shannon capacity () and fix any > 0. We construct, for any sufficiently small > 0, binary linear codes of block length (1/ 2+) and rate () − that enable reliable communication on 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 (1/ 2). This quadratic dependence on the gap 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. 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.

Polar Codes’ Simplicity, Random Codes’ Durability

“…(Quantization) Control the cardinality of the output alphabets of the synthetic channels [46], [64], [93]. This step is a necessity for application because real numbers are not realrounding errors emerges in the a posteriori probabilities in Section III.…”

“…Prudent readers are invited to check [30, the paragraph before Section III], [46], [50], [72], [75], [76] for a list of inhomogeneous configurations of kernels. See [30], [31], [77] for how nonlinear bijections are similar to (or different from) linear bijections.…”

“…What we just established is the successive cancellation decoder of polar codes that could be found in most works that implement polar codes. For instance, [ [46,Section 9] for an almost identical construction albeit they had q = 2 in mind. We replicate the whole story to demonstrate that each DU may use a unique bijection "g" without changing the overall structure too much.…”

“…large random kernels achieve 2ρ < 1 over BECs, breaking the barrier. Guruswami et al [46], [47] extended their result to all BDMCs utilizing the dynamic kernel technique. Over the remaining channels, the present work fills the gap.…”

“…This makes it a direct generalization of [48] to all polarizing kernel matrices G over all prime-ary channels. The preprint [47] contains a section that pushes the conference abstract [46] to positive π while maintaining ρ ≈ 1/2. Over the remaining channels, the present work fills the gap.…”