  We consider two problems related to polar codes. First is the problem of
polar codes construction and analysis of their performance without Monte-Carlo
method. The formulas proposed are the same as those in [Mori-Tanaka], yet we
believe that our approach is original and has clear advantages. The resulting
computational procedure is presented in a fast algorithm form which can be
easily implemented on a computer. Secondly, we present an original method of
construction of concatenated codes based on polar codes. We give an algorithm
for construction of such codes and present numerical experiments showing
significant performance improvement with respect to original polar codes
proposed by Ar\i kan. We use the term \emph{concatenated code} not in its
classical sense (e.g. [Forney]). However we believe that our usage is quite
appropriate for the exploited construction. Further, we solve the optimization
problem of choosing codes minimizing the block error of the whole concatenated
code under the constraint of its fixed rate.
