Schloss Dagstuhl – Leibniz-Zentrum für Informatik GmbH, Dagstuhl Publishing, Saarbrücken/Wadern,

Shannon’s monumental 1948 work laid the foundations for the rich fields of information andcoding theory. The quest for efficient coding schemes to approach Shannon capacity has occupiedresearchers ever since, with spectacular progress enabling the widespread use of error-correctingcodes in practice. Yet the theoretical problem of approaching capacity arbitrarily closely withpolynomial complexity remained open except in the special case of erasure channels.In 2008, Arikan proposed an insightful new method for constructing capacity-achieving codesbased on channel polarization. In this talk, I will begin with a self-contained survey of Arikan’scelebrated construction of polar codes, and then discuss our recent proof (with Patrick Xia) that,for all binary-input symmetric memoryless channels, polar codes enable reliable communicationat rates within > 0 of the Shannon capacity with block length (delay), construction complexity,and decoding complexity all bounded by a polynomial in the gap to capacity, i.e., by poly(1/ ).Polar coding gives the first explicit construction with rigorous proofs of all these properties;previous constructions were not known to achieve capacity with less than exp(1/ ) decodingcomplexity.We establish the capacity-achieving property of polar codes via a direct analysis of the under-lying martingale of conditional entropies, without relying on the martingale convergence theorem.This step gives rough polarization (noise levels for the good channels), which can then be ad-equately amplified by tracking the decay of the channel Bhattacharyya parameters. Our effectivebounds imply that polar codes can have block length bounded by poly(1/ ). We also show thatthe generator matrix of such polar codes can be constructed in polynomial time by algorithmicallycomputing an adequate approximation of the polarization process. 1998 ACM Subject Classification E.4 Coding and Information Theory, F.2 Analysis ofAlgorithms and Problem Complexity