Polar codes were recently introduced by Arıkan. They achieve the capacity of arbitrary symmetric 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 × 2 matrixˆ1 0 1 1˜. It was shown by Arıkan 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 Arıkan 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 first show that any ℓ × ℓ matrix none of whose column permutations is upper triangular polarizes symmetric channels. We then 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 n achieves exponents arbitrarily close to 1 and which exceeds 1 2 for size 16.

Polar codes, introduced recently by Arıkan, are the first family of codes known to achieve capacity of symmetric channels using a low complexity successive cancellation decoder. Although these codes, combined with successive cancellation, are optimal in this respect, their finite-length performance is not record breaking. We discuss several techniques through which their finite-length performance can be improved. We also study the performance of these codes in the context of source coding, both lossless and lossy, in the single-user context as well as for distributed applications.

Abstract-We consider lossy source compression of a binary symmetric source with Hamming distortion function. We show that polar codes combined with a low-complexity successive cancellation encoding algorithm achieve the rate-distortion bound. The complexity of both the encoding and the decoding algorithm is O (N log(N )), where N is the blocklength of the code. Our result mirrors Arıkan's capacity achieving polar code construction for channel coding.

Abstract-We consider lossy source compression of a binary symmetric source using polar codes and the low-complexity successive encoding algorithm. It was recently shown by Arıkan that polar codes achieve the capacity of arbitrary symmetric binary-input discrete memoryless channels under a successive decoding strategy. We show the equivalent result for lossy source compression, i.e., we show that this combination achieves the rate-distortion bound for a binary symmetric source. We further show the optimality of polar codes for various problems including the binary Wyner-Ziv and the binary Gelfand-Pinsker problem.