Strong Polarization for Shortened and Punctured Polar Codes
Boaz Shuval, Ido Tal
TL;DR
This work extends polar codes to arbitrary codelengths by applying shortening and puncturing transforms, enabling capacity-achieving communication without restricting to powers of two. It develops a generalized polar framework that uses extended transforms and special distributions to handle shortened and punctured bits, and proves that both transforms preserve polarization with the same exponential error decay as classical polar codes. The main contribution is a unified, general theorem showing that for any large enough length $M$, the polarized indices concentrate around the channel capacity $I(X;Y)$ and the conditional entropy $H(X|Y)$ with the same $2^{-M^{\beta}}$ decay, achievable with $O(M\log M)$ complexity. This has practical impact by enabling efficient, length-flexible polar coding schemes for binary-input memoryless channels, including non-symmetric cases, while retaining strong performance guarantees.
Abstract
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.