Puncturing and Shortening for Polar Codes via the Partial Order by Binary Domination

In this paper, we investigate the fundamentals of puncturing and shortening for polar codes, based on binary domination which plays a key role in polar code construction. We first prove that the orders of encoder input bits to be made incapable (by puncturing) or to be shortened are governed by binary domination. In particular, we show that binary domination completely determines incapable or shortened bit patterns for polar codes, and that all the possible incapable or shortened bit patterns can be identified. We then present the patterns of the corresponding encoder output bits to be punctured or fixed, when the incapable or shortened bits are given.