Improved bounds on the finite length scaling of polar codes

Improved upper bounds on the blocklength required to communicate over binary-input channels using polar codes, below some given error probability, are derived. For that purpose, an improved bound on the number of non-polarizing channels is obtained. The main result is that the blocklength required to communicate reliably scales at most as (O((I(W)-R^-5.702})) , where (R) is the code rate and (I(W)) is the symmetric capacity of the channel (W). The results are then extended to polar lossy source coding at rate (R) of a source with symmetric distortion-rate function (Dc)). The blocklength required scales at most as (O((D^0-5.702})) , where (D⁰) is the maximal allowed gap between the actual average (or typical) distortion and (D(R)).