  Gaussian approximation (GA) is widely used to construct polar codes. However
when the code length is long, the subchannel selection inaccuracy due to the
calculation error of conventional approximate GA (AGA), which uses a
two-segment approximation function, results in a catastrophic performance loss.
In this paper, new principles to design the GA approximation functions for
polar codes are proposed. First, we introduce the concepts of polarization
violation set (PVS) and polarization reversal set (PRS) to explain the
essential reasons that the conventional AGA scheme cannot work well for the
long-length polar code construction. In fact, these two sets will lead to the
rank error of subsequent subchannels, which means the orders of subchannels are
misaligned, which is a severe problem for polar code construction. Second, we
propose a new metric, named cumulative-logarithmic error (CLE), to
quantitatively evaluate the remainder approximation error of AGA in logarithm.
We derive the upper bound of CLE to simplify its calculation. Finally, guided
by PVS, PRS and CLE bound analysis, we propose new construction rules based on
a multi-segment approximation function, which obviously improve the calculation
accuracy of AGA so as to ensure the excellent performance of polar codes
especially for the long code lengths. Numerical and simulation results indicate
that the proposed AGA schemes are critical to construct the high-performance
polar codes.
