  We present a comparative study of the performance of various polar code
constructions in an additive white Gaussian noise (AWGN) channel. A polar code
construction is any algorithm that selects $K$ best among $N$ possible polar
bit-channels at the design signal-to-noise-ratio (design-SNR) in terms of bit
error rate (BER). Optimal polar code construction is hard and therefore many
suboptimal polar code constructions have been proposed at different
computational complexities. Polar codes are also non-universal meaning the code
changes significantly with the design-SNR. However, it is not known which
construction algorithm at what design-SNR constructs the best polar codes. We
first present a comprehensive survey of all the well-known polar code
constructions along with their full implementations. We then propose a
heuristic algorithm to find the best design-SNR for constructing best possible
polar codes from a given construction algorithm. The proposed algorithm
involves a search among several possible design-SNRs. We finally use our
algorithm to perform a comparison of different construction algorithms using
extensive simulations. We find that all polar code construction algorithms
generate equally good polar codes in an AWGN channel, if the design-SNR is
optimized.
