Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels

• Published in: IEEE transactions on information theory Vol. 55; no. 7; pp. 3051 - 3073
• Main Author: Arikan, Erdal
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.07.2009; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Binary system; Blocking; Cancellation; Capacity planning; Capacity-achieving codes; Channel capacity; Channel coding; channel polarization; Channels; Codes; Coding, codes; Communication channels; Construction; Councils; Decoding; Encoders; Exact sciences and technology; Information theory; Information, signal and communications theory; Memoryless systems; Noise cancellation; Plotkin construction; polar codes; Polarization; Reed- Muller (RM) codes; Reproduction; Signal and communications theory; successive cancellation decoding; Systems, networks and services of telecommunications; Telecommunications; Telecommunications and information theory; Transmission and modulation (techniques and equipments); Performance evaluation; Binary channel; Channel capacity; Discrete channel; successive cancellation decoding; Information rate; n channel; Decoding; Channel coding; Information transmission; Memoryless channel; Codec; Capacity-achieving codes; Algorithm complexity; Reed-Muller (RM) codes; Plotkin construction; polar codes; channel polarization; Reed Muller code
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2009.2021379

A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity I(W) of any given binary-input discrete memoryless channel (B-DMC) W . The symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability. Channel polarization refers to the fact that it is possible to synthesize, out of N independent copies of a given B-DMC W , a second set of N binary-input channels \{W_N^{(i)}:1\le i\le N\} such that, as N becomes large, the fraction of indices i for which I(W_N^{(i)}) is near 1 approaches I(W) and the fraction for which I(W_N^{(i)}) is near 0 approaches 1-I(W) . The polarized channels \{W_N^{(i)}\} are well-conditioned for channel coding: one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining. Codes constructed on the basis of this idea are called polar codes. The paper proves that, given any B-DMC W with I(W)> 0 and any target rate R ≪ I(W) , there exists a sequence of polar codes \{{\Fraktur {C}}_n;n\ge 1\} such that {\Fraktur {C}}_n has block-length N=2^n , rate \ge R , and probability of block error under successive cancellation decoding bounded as P_{e}(N,R) \le O(N^{-{1\over 4}}) independently of the code rate. This performance is achievable by encoders and decoders with complexity O(N\log N) for each.