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 scheme. 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 0 1 1]. 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 blocklength. It was already mentioned by Arikan that in principle larger matrices can be used to construct polar codes. In this paper, it is first shown that any l x l matrix none of whose column permutations is upper triangular polarizes binary-input memoryless channels. The exponent of a given square matrix is characterized, upper and lower bounds on achievable exponents are given. Using these bounds it is shown that there are no matrices of size smaller than 15 x 15 with exponents exceeding 1/2. Further, a general construction based on BCH codes which for large l achieves exponents arbitrarily close to 1 is given. At size 16 x 16, this construction yields an exponent greater than 1/2.
polar codes polar code construction error exponent channel polarization rate of polarization