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 D(·). The blocklength required scales at most as O((D 0 ) -5.702 ), where D 0 is the maximal allowed gap between the actual average (or typical) distortion and D(R).