  Polar codes are a family of capacity-achieving codes that have explicit and
low-complexity construction, encoding, and decoding algorithms. Decoding of
polar codes is based on the successive-cancellation decoder, which decodes in a
bit- wise manner. A decoding error occurs when at least one bit is erroneously
decoded. The various codeword bits are correlated, yet performance analysis of
polar codes ignores this dependence: the upper bound is based on the union
bound, and the lower bound is based on the worst-performing bit. Improvement of
the lower bound is afforded by considering error probabilities of two bits
simultaneously. These are difficult to compute explicitly due to the large
alphabet size inherent to polar codes. In this research we propose a method to
lower-bound the error probabilities of bit pairs. We develop several
transformations on pairs of synthetic channels that make the resultant
synthetic channels amenable to alphabet reduction. Our method yields lower
bounds that significantly improve upon currently known lower bounds for polar
codes under successive-cancellation decoding.
