  The definition of polar codes given by Arikan is explicit, but the
construction complexity is an issue. This is due to the exponential growth in
the size of the output alphabet of the bit-channels as the codeword length
increases. Tal and Vardy recently presented a method for constructing polar
codes which controls this growth. They approximated each bit-channel with a
better channel and a worse channel while reducing the alphabet size. They
constructed a polar code based on the worse channel and used the better channel
to measure the distance from the optimal channel. This paper considers the
knowledge gained from the perspective of the better channel. A method is
presented using iterative upgrading of the bit-channels which successively
results in a channel closer to the original one. It is shown that this approach
can be used to obtain a channel arbitrarily close to the original channel, and
therefore to the optimal construction of a polar code.
