  In this article, we propose the construction of polar codes based on
piecewise Gaussian approximation (PGA) techniques. The PGA is first optimized
and then compared to the Gaussian approximation (GA) construction method,
showing performance gains for medium blocks and high precision for long blocks,
in scenarios with successive cancellation (SC) decoding and additive white
gaussian noise (AWGN) channel. Based on the PGA, we develop two approximations
based on multi-segmented polynomials that are easy to implement. We present the
Approximate PGA (APGA) that is optimized for medium blocks and provides a
performance improvement without increasing complexity. Furthermore, we develop
the simplified PGA (SPGA) as an alternative to the GA, which is optimized for
long blocks and achieves high construction accuracy. Simulation results show
that the APGA and SPGA construction methods outperform existing GA and
competing approaches for medium and long block codes with notable performance
improvement.
