On the Analysis of Puncturing for Finite-Length Polar Codes: Boolean Function Approach

This paper investigates the impact of puncturing on finite-length polar codes in which a puncturing pattern p = (p0, ..., pN ) is applied to a length-N polar code.. We first introduce two virtual channels to stochastically model the punctured (untransmitted) bits, which are respectively called useless channel model (UCM) and deterministic channel model (DCM). Under each model, we derive boolean functions in variables p0, ..., pN−1 that can indicate which polarized channels should carry frozen bits. Based on this, we present an efficient method to jointly optimize a puncturing pattern and an information set. Focusing on a fixed information set, we show that there exist the so-called catastrophic puncturing patterns that will surely lead to a block error and derive their weight distributions recursively. We then propose the two construction methods of a rate-compatible (RC) polar code which ensures that each puncturing pattern in the family is non-catastrophic. Simulation results demonstrate that the proposed RC polar code outperform the RC Turbo code adopted in LTE. January 17, 2018 DRAFT ar X iv :1 80 1. 05 09 5v 1 [ cs .I T ] 1 6 Ja n 20 18