English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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. Our codes are a variant of Arikan’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.

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