  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. We also demonstrate that the
order and the pattern of puncturing and shortening for polar codes can be
aligned. In the previous work on the rate matching for polar codes, puncturing
of encoder output bits begins from a low-indexed bit, while shortening starts
from a high-indexed bit. Unlike such a conventional approach, we show that
encoder output bits can be punctured from high-indexed bits, while keeping the
incapable bit pattern exactly the same. This makes it possible to design a
unified circular-buffer rate matching (CB-RM) scheme that includes puncturing,
shortening, and repetition.
