低密度奇偶檢查碼(LDPC code)在1962年被Gallager提出，其錯誤校正能力非常接近理論最大值(Shannon Limit),其中準循環低密度奇偶檢查碼(quasi-cyclic LDPC code)為具有碼字半循環特性的更正碼，此特性提供了編碼及解碼上複雜度降低的優勢。
本論文提出一種基於矩陣擴展的準循環低密度奇偶檢查碼建構演算法，這套演算法是建立於PEG(progress edge growth)演算法循序建構 Tanner 圖中連線的設計方式。
我們透過PEG演算法建立出小尺寸的基矩陣，再藉由深度樹(depth tree)的延伸找出最適合的矩陣擴展參數。程式模擬結果顯示我們的演算法在相同的設計變數下能達到不錯的更正能力。

The error-correcting code (ECC) is one of the channel coding technique widely used in many applications.
Low density parity check (LDPC) codes have a remarkable performance Among the error-correcting codes, and have drawn significant attention for their error-correcting capability using the message-passing decoding algorithm.
The PEG algorithm is a good sub-optimal method to construct the Tanner graph for a LDPC code. By progressively establishing edges between the furthest nodes, the PEG algorithm is able to build a Tanner graph while maximizing length of the cycle caused in the edge adding progress.
In this thesis, we introduces a greedy algorithm for quasi-cyclic (QC) LDPC code construction, which works in the same manner as the PEG algorithm. QC-LDPC codes are a subclass of LDPC codes which gains benefits on encoding/decodng complexity reduction from their circular shifting structures.
In the proposed algorithm, a QC-LDPC code is built by extending a small size matrix to the target size with large cycles. Our algorithm focuses on cycle length stretching during the matrix extension.
The matrix extension is implemented by fitting proper circulant matrix while eliminating short cycles. We present a new kind of depth tree for circulant search in matrix extension procedure.
Our algorithm brings out the QC-LDPC code with reconfigurable code parameter settings and has good performance close to PEG LDPC codes. Furthermore, the use of matrix extension reduces the computation scale, and provides a efficient and easy-to-implement method to construct a QC-LDPC code.

Abstract i
Contents ii
1 Introduction 1
2 System Description 3
2.1 LDPC Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 QC-LDPC Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Encoding LDPC codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Decoding Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Code Construction Algorithms 11
3.1 Progressive Edge-Growth Algorithm . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 QC-LDPC Progressive Edge-Growth Algorithm . . . . . . . . . . . . . . . . 14
3.3 Proposed QC-LDPC Extension Algorithm Based on Depth Tree . . . . . . . 16
4 Simulations 21
4.1 simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Complexity Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5 Conclusion 25

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