On the scaling of the blocklength in some polar coding schemes

Polar code-based source coding is considered for some given code rate and some desired distortion level. The distortion level of a typical quantized source vector is required to be below the desired distortion level. A sufficient polynomial scaling law of the blocklength with respect to the gap between the desired distortion and the distortion-rate function is derived. The results are then used to obtain a polynomial scaling of the blocklength with respect to the gap to capacity for the polar code-based solution of the binary dirty paper problem and for polar write-once memory codes.