Sonlu uzunluklu kutupsal kodlar için üretim matrisi seçimi

Polar coding is a recently proposed coding technique, which has been proven to achieve the channel capacity. The original 2x2 generator matrix polarizes the channels and a portion of channels' capacity approach 1, while the remaining channel capacities approach 0. In the literature, it was shown that, as the codelength goes to infinity, polarization performance of Arıkan's 2x2 matrix is better than any matrix of size less than 16x16. In this thesis, we show that this observation does not necessarily hold for the finite-length case and the channel polarization is attainable by using different generator matrices. The main contribution of this thesis consists of filling the gap on the analysis of the finite-length polar code generation. A normalized polarization distance measure was defined and polar codes from different generator matrices showing different amount of polarization effects were obtained using this measure. Also, the coding structure for these generalized polar codes were obtained. Polarization performances in both asymptotical and finite-length cases were investigated especially for generator matrices of size 3x3 and 4x4 using Bhattacharyya parameter histograms, polarization rate exponents and normalized polarization distance measures; also upper bound on block error probabilities for these matrices were analyzed. Moreover, the recursive likelihood ratio equations for a specific 4x4 matrix showing the best polarization performance among all 4x4 generator matrices were defined. A decoding algorithm was implemented for a generator matrix from the best group of 4x4 generator matrices and its erasure rate was compared with the Arıkan's original generator matrix' decoding performance.