Abstract
Polar Ìý codes Ìýwere 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 originalÌý polar ÌýcodeÌý construction Ìýis closely related to the recursiveÌý construction Ìýof Reed-MullerÌý codes Ìýand is based on the 2 × 2 matrix of the given equation. It was shown by Arikan and Telatar that thisÌý construction Ìýachieves an errorÌý exponent Ìýof 1/2, i.e., that for sufficiently large block-lengths 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 constructÌý polar Ìý codes . A fundamental question then is to see whether there exist matrices withÌý exponent Ìýexceeding 1/2. We characterize theÌý exponent Ìýof a given square matrix and derive upper and lowerÌý bounds Ìýon achievableÌý exponents . Using theseÌý bounds Ìýwe show that there are no matrices of size less than 15 withÌý exponents Ìýexceeding 1/2. Further, we give a generalÌý construction Ìýbased on BCHÌý codes Ìýwhich for large matrix sizes achievesÌý exponents Ìýarbitrarily close to 1 and which exceeds 1/2 for size 16.