Polar Codes: Bounds on Bhattacharyya Parameters and Their Applications

This paper proposes a novel lower bound on the Bhattacharyya parameter of synthesized channels in polar codes. The proposed bound is a result of a newly defined merge operation that reduces the cardinality of output alphabets of a symmetric binary-input discrete memoryless channel. By repeated applications of the proposed merge operation, we establish a universal bound on the Bhattacharyya parameters of synthesized channels that holds for any symmetric binary-input discrete memoryless transmission channel. We then discuss some applications of the new bound. For example, using the derived universal lower bound along with the existing upper bound, we find a partial order among the Bhattacharyya parameters of those synthesized channels that cannot be ordered through existing results. This ordering can help efficient design of polar codes. In addition, some extremal behaviors of the binary-symmetric channel are also proved.