有限域已經被廣泛的應用於強大的準循環低密度奇偶校驗碼(QC-LDPC code)的建構中，儘管如此，因為有限域的性質，所建構的碼長度是相當有限的。例如本論文所介紹的用兩個最小重RS碼字的建構法要先建構出有限域，有限域又是由質數或質數次方所建，最後建構出的準循環低密度奇偶校驗碼的擴展矩陣大小就會是質數減一或是質數次方減一，碼長度也會因此而受限。為了能建構更彈性的碼長度，本論文提出了兩種新的建構方式。第一種是基於仿射變換(affine transformation)，主要是利用仿射變換公式建構出基本矩陣(base matrix)，再用其特性證明建構出來的準循環低密度奇偶校驗碼不會有短環4(cycle 4)，第二種是方法是利用電腦輔助搜尋，此方式是從RS建構法中觀察其基本矩陣，發現基本矩陣為循環矩陣，接著短環4公式的相關運算通常為質數減一或質數次方減一而非質數，所以先建構一循環矩陣，再利用循環的特性找出所有出現短環4的地方並將其消除，最終得到一個擴展矩陣大小不為質數所限制的基本矩陣。用這兩種方法所得到的準循環低密度奇偶校驗碼將不會有短環4。利用所提出的方法，將能建構出在錯誤率上有競爭力且有更加彈性碼長度的準循環低密度奇偶校驗碼，但是這兩種建構方式也存在著基本矩陣太小的共同缺點，所以其應用有侷限性。

Finite fields have been widely used for the constructions of powerful quasi-cyclic (QC) low-density parity-check (LDPC) codes. However, due to the nature of the finite fields, the lengths of the resultant codes are quite limited. In order to achieve more flexible code lengths, two novel code construction methods are proposed in this thesis. The first is based on the affine transformation. The second is assisted by the computer search. Using these two methods, the Tanner graphs of the resultant QC-LDPC codes are cycle-4 free. Using the proposed methods, QC-LDPC codes which can achieve a competitive error–rate performance can be easily constructed with much more flexible code lengths.

1 介紹1
2 基於有限域的準循環低密度奇偶校驗碼的建構3
2.1 矩陣的擴展-1
2.2 準循環低密度奇偶校驗碼的通用建構法
2.3 用兩個最小重RS 碼字的建構法
2.4 矩陣的擴展-2
2.5 短環的介紹
3 非有限域的建構方式13
3.1 仿射變換的建構方式
3.2 電腦輔助建構方式
3.2.1 高碼率改善方式
3.3 模擬結果
4 建碼的其他考量
4.1 二級編碼
4.2 掩蔽
5 總結39
6 附錄41
6.1 短環4的證明
6.2 改變擴展矩陣大小的方式
6.2.1 放大擴展矩陣
6.2.2 縮小擴展矩陣

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