在此篇論文，我們應用非均等保護碼( unequal error protection codes) 於極性碼(polar codes)的接續消去名單解碼法(successive cancellation list decoding)上。

此方法是利用刪除非均等保護碼之奇偶檢查矩陣(parity check matrix)的非基底的行來得到不同碼率(code rate)的非均等保護碼。非均等保護碼可以有效率的編碼方式對碼字(codeword)前段的訊息位元(information bits)做奇偶檢查。我們使用非均等保護碼的奇偶檢查矩陣來幫助檢查接續消去名單解碼法，協助判斷被估測的部分碼字是否能繼續留在名單內。
模擬結果顯示，極性碼結合非均等保護碼通常比極性碼結合多重嵌套循環同位檢查(multiple nested cyclic redundancy check)的效果好。我們也顯示了在低碼率情形下，極性碼結合非均等保護碼的收斂情況會比高碼率來的快。在第一章我們介紹極性碼的作者Arikan，接著分析過去提出的不同種極性碼解碼法，在第二章我們介紹極性碼的原理與極性碼的數學模型與重要參數，並且解釋不同種極性碼解碼法的演算法，在第三章我們介紹非均等保護碼的定義與性質，說明何謂最佳非均等保護碼，同時介紹兩種最佳非均等保護碼的建構方法，最後我們透過數學推導出的三層非均等保護碼，在第五章的模擬將使用以上介紹的三種非均等保護碼，在第四章我們結合極性碼與非均等保護碼，透過分析極性碼的極化後的不同通道，來放置奇偶檢查在特別位置能得到更好的錯誤率，在第五章我們顯示不同參數下的圖，最後在第六章給予本篇論文的結論。

In this thesis, we use unequal error protection (UEP) codes in the successive
cancellation list decoding (SCLD) of polar codes. We puncture non-basis columns
of the parity check matrix of an optimal UEP code to get different rates of UEP
codes. UEP code can do parity check for the front part of information bits
of codeword by using an eifficient encoding method. Then we use this parity
check matrix of the UEP code to check paths which are valid. Simulation results
show that polar codes concatenated with UEP codes generally outperform polar
codes concatenated with multiple CRCs. And we also show that the saturated
phenomenon of low rate case is faster than high rate case.

1 Introduction 4
2 Polar Codes 5
2.1 Denitions and Properties . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Channel Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Channel combining . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Channel splitting . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.3 Iterative formulas of channel transformations . . . . . . . . 8
2.2.4 Channel polarization . . . . . . . . . . . . . . . . . . . . . 9
2.3 Code Construction for BECs and BSCs . . . . . . . . . . . . . . 10
2.4 Encoding of Polar Codes . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Decoding of the Polar Codes . . . . . . . . . . . . . . . . . . . . . 11
2.5.1 Successive cancellation decoding . . . . . . . . . . . . . . . 11
2.5.2 Successive cancellation list decoding . . . . . . . . . . . . . 11
2.5.3 Successive cancellation list decoding with a single CRC . . 12
2.5.4 Successive cancellation list decoding with multiple CRC . . 13
3 Linear Unequal Error Protection Codes 14
3.1 Denitions and Properties . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Optimal Linear Binary Systematic UEP Codes . . . . . . . . . . . 17
3.3 Method of Constructing Optimal Linear Binary Systematic UEP
Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 UEP Concatenated Polar Codes 22
4.1 UEP Concatenated Polar Codes . . . . . . . . . . . . . . . . . . . 22
4.2 UEP Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Successive Cancellation List Decoding of Polar Codes with UEP . 23
4.4 UEP Check Position Comparison . . . . . . . . . . . . . . . . . . 25
4.5 Saturated Phenomenon Discussion . . . . . . . . . . . . . . . . . 25
5 Simulations 30
5.1 Simulation Settings . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.2.1 N=128, L=2, rate=0.5, UEP Lemma 6, r=15 . . . . . . . 34
5.2.2 N=128, L=4, rate=0.5, UEP Lemma 6, r=15 . . . . . . . 35
5.2.3 N=128, L=8, rate=0.5, UEP Lemma 6, r=15 . . . . . . . 36
5.2.4 N=128, L=16, rate=0.5, UEP Lemma 6, r=15 . . . . . . . 37
5.2.5 N=128, L=32, rate=0.5, UEP Lemma 6, r=15 . . . . . . . 38
5.2.6 N=128, L=2, rate=0.5, UEP Lemma 6 & Lemma 7 comparison,
r=15, r=20 . . . . . . . . . . . . . . . . . . . . . . 40
5.2.7 N=128, L=4, rate=0.5, UEP Lemma 6 & Lemma 7 comparison,
r=15, r=20 . . . . . . . . . . . . . . . . . . . . . . 41
5.2.8 N=128, L=8, rate=0.5, UEP Lemma 6 & Lemma 7 comparison,
r=15, r=20 . . . . . . . . . . . . . . . . . . . . . . 42
5.2.9 N=128, L=16, rate=0.5, UEP Lemma 6 & Lemma 7 comparison,
r=15, r=20 . . . . . . . . . . . . . . . . . . . . . . 43
5.2.10 N=256, L=2, rate=0.5, UEP Lemma 7, r=20 . . . . . . . 45
5.2.11 N=256, L=4, rate=0.5, UEP Lemma 7, r=20 . . . . . . . 46
5.2.12 N=256, L=8, rate=0.5, UEP Lemma 7, r=20 . . . . . . . 47
5.2.13 N=256, L=16, rate=0.5, UEP Lemma 7, r=20 . . . . . . . 48
5.2.14 N=256, L=2, rate=0.5, UEP Theorem 1, r=30 . . . . . . . 50
5.2.15 N=256, L=4, rate=0.5, UEP Theorem 1, r=30 . . . . . . . 51
5.2.16 N=256, L=8, rate=0.5, UEP Theorem 1, r=30 . . . . . . . 52
5.2.17 N=256, L=16, rate=0.5, UEP Theorem 1, r=30 . . . . . . 53
5.2.18 N=256, L=2, rate=0.6, UEP check position comparison,
r=15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2.19 N=256, L=4, rate=0.6, UEP check position comparison,
r=15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2.20 N=256, L=4, rate=0.5, UEP Lemma 6, Lemma 7, Theorem
1 comparison, r=15, 20, 30 . . . . . . . . . . . . . . . 56
5.2.21 N=256, L=16, rate=0.5, UEP Lemma 6, Lemma 7, Theorem
1 comparison, r=15, 20, 30 . . . . . . . . . . . . . . . 57
5.2.22 N=256, rate=0.6, UEP Theorem 1, L=8, 32, 128 comparison,
r=30 . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2.23 N=128, rate=0.6, UEP Lemma 6, L=32, 128, 512, 1024
comparison, r=15 . . . . . . . . . . . . . . . . . . . . . . . 61
6 Conclusion 62

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