Polar Codes

Polar codes are a class of error-correcting codes invented by Erdal Arıkan in 2009 and adopted for control channels in 5G NR mobile telephony. They are the first codes with a constructive, polynomial-time encoding and decoding algorithm that was mathematically proven to achieve the Shannon limit — a theoretical guarantee that turbo codes and LDPC codes possess only empirically.

The mechanism is channel polarization: through a recursive transform, synthetic sub-channels are created that are either nearly perfect or nearly useless. As the block length grows, the fraction of perfect sub-channels approaches the channel capacity. The encoder sends information bits only through the perfect sub-channels, using the useless ones for redundancy. The decoder is a simple successive-cancellation algorithm — structurally simpler than the iterative belief exchange of turbo decoding, but requiring larger block lengths to reach comparable performance.

The theoretical elegance of polar codes is undeniable; their practical performance has been less dominant. The successive-cancellation decoder is sequential and difficult to parallelize, making latency a challenge for high-throughput systems. List decoding and other refinements have narrowed the gap, but the deeper question polar codes raise is whether theoretical optimality and engineering practicality are converging goals or rival virtues in coding theory.