  This paper introduces a class of specific puncturing patterns, called
symmetric puncturing patterns, which can be characterized and generated from
the rows of the generator matrix $G_N$. They are first shown to be
non-equivalent, then a low-complexity method to generate symmetric puncturing
patterns is proposed, which performs a search tree algorithm with limited
depth, over the rows of $G_N$. Symmetric patterns are further optimized by
density evolution, and shown to yield better performance than state-of-the-art
rate compatible code constructions, relying on either puncturing or shortening
techniques.
