Polar codes: Characterization of exponent, bounds, and constructions

Polar codes were recently introduced by Arikan. They achieve the symmetric capacity of arbitrary binary-input discrete memoryless channels under a low complexity successive cancellation decoding strategy. The original polar code construction is closely related to the recursive construction of Reed-Muller codes and is based on the 2 x 2 matrix [(1) (1)(1) (0)]. It was shown by Arikan and Telatar that this construction achieves an error exponent of 1/2 i.e., that for sufficiently large blocklengths the error probability decays exponentially in the square root of the length. It was already mentioned by Arikan that in principle larger matrices can be used to construct polar codes. A fundamental question then is to see whether there exist matrices with exponent exceeding 1/2. We characterize the exponent of a given square matrix and derive upper and lower bounds on achievable exponents. Using these bounds we show that there are no matrices of size less than 15 with exponents exceeding 1/2. Further, we give a general construction based on BCH codes which for large matrix sizes achieves exponents arbitrarily close to 1 and which exceeds 1/2 for size 16.


Published in:
Proceedings of the IEEE International Symposium on Information Theory, 1483-1487
Presented at:
IEEE International Symposium on Information Theory ISIT 2009, Seoul, June 28 - July 3, 2009
Year:
2009
Publisher:
Ieee Service Center, 445 Hoes Lane, Po Box 1331, Piscataway, Nj 08855-1331 Usa
Laboratories:




 Record created 2009-11-30, last modified 2018-03-17


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