本論文提出一種強化極性碼(polar codes)中的翻轉位元演算法(bit-flipping algorithm)所需要的臨界集合(critical set)。在文獻中，其中一種建立方式藉由極 化碼的通道極化(channel polariztion)的性質估計出極化位元通道的錯誤率，進 行翻轉位元演算法時，由錯誤率最高到最低的位元依序做位元翻轉。在本論文 加入信心傳遞(belief propagation)解碼的結果做向後追蹤(trace back)，在第一次 的信心傳遞解碼後，在每一層的檢查節點(check node)有不滿足的情況下，向 後回推相連的變數節點(variable node)，並記錄變數節點的位置，一層層地回推 變數節點的位置找出可能地錯誤的位元位置，利用兩種方式所產生的臨界集合 合併考慮建立更為精準的臨界集合做翻轉位元演算法。並從模擬結果發現，此 方法可以有效地降低解碼器的錯誤率以及減少翻轉位元演算法所花費的翻轉次 數以及解碼時間。

In this thesis, we propose a new strategy to construct a critical set needed to enhance the bit-flipping algorithm in the decoding of polar codes. In the literature, one of the methods to establish a critical set is to estimate a priori error rates of polarized bit channels by using the channel polarization property of polar codes. However, we combine this error rates of estimation of polarized bit channels and the trace back method in the literation after the first belief propagation decoding, where, if the check node in one layer is not satisfied, the check node traces back to the connected variable nodes, and then records the position of the variable nodes layer by layer to find out the possible wrong bit positions. By using our proposed method, the merged critical set is more accurate to target error prone bits to be flipped in the bit-flipping algorithm. From the simulation results, it can be seen that our proposed method can effectively reduce the block error rate as well as the number of attempts and decoding time in the bit-flipping algorithm.

Contents
1 Introduction 1
2 Polar Codes 3
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Capacity and Reliability of DMCs . . . . . . . . . . . . . . . . . . . 4
2.3 Kronecker Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Channel Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4.1 Channel Combining . . . . . . . . . . . . . . . . . . . . . . 10
2.4.2 Channel Splitting . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Polar Codes Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Polar Codes Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6.1 Belief Propagation . . . . . . . . . . . . . . . . . . . . . . . 21
3 Bit-Flipping Decoders 28
3.1 Critical Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Error Rates of Polarized Bit Channels . . . . . . . . . . . . . . . . . 31
3.3 Trace Back Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 The Merged Critical Set . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5 Bit-Flipping Decoders . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Simulations 42
4.1 Simulation Configurations . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5 Conclusion 49
Bibliography 49
A Experimental data 52

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