Polar codes: Reliable communication with complexity polynomial in the gap to Shannon capacity

November 13, 2013

Shannon's monumental 1948 work laid the foundations for the rich fields of information and coding theory. The quest for *efficient* coding schemes to approach Shannon capacity has occupied researchers ever since, with spectacular progress enabling the widespread use of error-correcting codes in practice. Yet the theoretical problem of approaching capacity arbitrarily closely with polynomial complexity remained open except in the special case of erasure channels.

In 2008, Arikan proposed an insightful new method for constructing capacity-achieving codes based on channel polarization. In this talk, I will begin with a brief survey of Arikan's celebrated construction of polar codes, and then discuss our proof (with Patrick Xia) that, for all binary-input symmetric memoryless channels, polar codes enable reliable communication at rates within epsilon > 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/epsilon). 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/epsilon) decoding complexity.

We establish the capacity-achieving property of polar codes via a direct analysis of the underlying martingale of conditional entropies, without relying on the martingale convergence theorem. This step gives rough polarization (noise levels ~ epsilon for the "good channels"), which can then be adequately amplified by tracking the decay of the channel Bhattacharyya parameters. Our effective bounds imply that polar codes can have block length bounded by poly(1/epsilon). The generator matrix of such polar codes can be constructed in polynomial time by algorithmically computing an adequate approximation of the polarization process.

In 2008, Arikan proposed an insightful new method for constructing capacity-achieving codes based on channel polarization. In this talk, I will begin with a brief survey of Arikan's celebrated construction of polar codes, and then discuss our proof (with Patrick Xia) that, for all binary-input symmetric memoryless channels, polar codes enable reliable communication at rates within epsilon > 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/epsilon). 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/epsilon) decoding complexity.

We establish the capacity-achieving property of polar codes via a direct analysis of the underlying martingale of conditional entropies, without relying on the martingale convergence theorem. This step gives rough polarization (noise levels ~ epsilon for the "good channels"), which can then be adequately amplified by tracking the decay of the channel Bhattacharyya parameters. Our effective bounds imply that polar codes can have block length bounded by poly(1/epsilon). The generator matrix of such polar codes can be constructed in polynomial time by algorithmically computing an adequate approximation of the polarization process.