Polar codes, invented by Erdal Arikan in 2008, are error correcting codes that are based on an entirely new principle and have many desirable properties. They are inherently of low complexity both for encoding and decoding, their analysis is simple, they allow to achieve capacity, and the underlying idea is broad and flexible and can hence be applied to a variety of problems.

Starting from scratch, we will see how simple notions of information theory form the basis for the polarization phenomenon, how these codes can be constructed and decoded efficiently, how they perform, and how to extend the basic idea to more complex scenarios.

Lecture 1: The Polarization Phenomenon

• binary erasure channel: proof of convergence
• binary-input memoryless output-symmetric channels: Mrs. Gerber’s lemma and proof of convergence

Lecture 2: Polar Codes

• basic scheme
• successive decoder
• upper bound on error probability
• efficient construction
• universality and partial orders
• list decoding for improved performance

Lecture 3: Error Exponent, Finite-Length Scaling, and Error Floor

• error exponent: tradeoff between error probability and block length at fixed channel
• finite-length scaling: tradeoff between channel parameter and block length at fixed error probability
• proof that polar codes have no error floor

Lecture 4: Extensions

• universal construction
• general kernels
• non-binary
• multi-terminal problems
• source codingraphy

To date there is a large number of resources on the topic of polar codes. Listed below are some links to videos, slides, and papers that you might find useful.

A Theory of Polarization, Emre Telatar, Simons Institute, Feb. 10, 2015