Strong Polarization for Shortened and Punctured Polar Codes

Polar codes were originally specified for codelengths that are powers of two. In many applications, it is desired to have a code that is not restricted to such lengths. Two common strategies of modifying the length of a code are shortening and puncturing. Simple and explicit schemes for shortening and puncturing were introduced by Wang and Liu, and by Niu, Chen, and Lin, respectively. In this paper, we prove that both schemes yield polar codes that are capacity achieving. Moreover, the probability of error for both the shortened and the punctured polar codes decreases to zero at the same exponential rate as seminal polar codes. These claims hold for \emph{all} codelengths large enough.