低密度偶校碼(Low-Density Parity-Check Codes)已經被證實當運用疊代訊號傳遞解碼及配合極長的碼長時，能有接近Shannon bound的表現。類循環低密度偶校碼(Quasi-Cyclic Low-Density Parity-Check Codes)則是現在常用的低密度偶校碼之一，他除了繼承低密度偶校碼的優點之外，其偶校矩陣還具有良好的代數結構，在硬體實作上能大幅降低製作的複雜度。因此，如何造出良好的類循環低密度偶校碼是近年來的重點之一。
在實際應用上，為了確定我們使用的低密度偶校碼能保護多少位元的資料，低密度偶校碼的碼率是我們必須知道的。一般來說，碼率可由偶校矩陣的行數減掉偶校矩陣的秩數來獲得，所以我們只要知道偶校矩陣的秩數就等同知道碼率。但是，對於大部分好的類循環低密度偶校碼來說，它們的偶校矩陣都不是滿秩數，所以會需要其他方法來計算矩陣的秩數。
在這篇論文裡，我們將循環矩陣用特徵多項式來表達，並以此發明了一些新方法來計算類循環低密度偶校碼的偶校矩陣的秩數。對於只有一塊列區塊的矩陣，我們找出了其秩數的公式解，並將這個結果推廣到有二或三塊列區塊的矩陣。對於其他無法用公式解的矩陣，我們發展了一套演算法，這套演算法可以處理所有這類型的偶校矩陣。最後，我們找了一些例子來說明我們的結果。

Low-density parity-check (LDPC) codes have been shown to have near-capacity performance with iterative message-passing decoding and sufficiently long block length. Quasi-cyclic LDPC (QC-LDPC) codes are an important class of LDPC codes which can be encoded and decoded with low complexity and suitable for many applications. In practical systems, it is important to know the exact dimension of a QC-LDPC code since the dimension describes the number of protected information bits. Since the dimension is equal to the code length minus the rank of the parity-check matrix, finding the code dimension is equivalent to finding the rank of the parity-check matrix. Unfortunately, the parity check matrix for QC-LDPC codes is usually not full-rank, so we need some methods to compute the rank.
In this thesis, we develop a new approach to calculate the exact rank of parity-check matrix for QC-LDPC codes based on the characteristic polynomials for circulant matrices.
A formula for the rank of the parity-check matrices with only one row-block is first derived.
We then extend the result to matrices with two or three row-blocks. These formulas are derived based on the technique that transforms the matrices into the upper-triangular form. Furthermore, a low-complexity algorithm is given to handle other types of matrices. Based
on the obtained results, the rank of the parity-check matrix for QC-LDPC codes under good algebraic constructions can be easily obtained. We also demonstrate our results by some examples.

Abstract i
Contents ii
1 Introduction 1
2 Low-Density Parity-Check Codes 3
2.1 Denition for LDPC Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Tanner Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Decoding Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Quasi-Cyclic Low-Density Parity-Check Codes 9
3.1 Denition for QC-LDPC Codes . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Algebraic Representations for Parity-Check Matrix . . . . . . . . . . . . . . 10
3.3 Some Constructions for QC-LDPC Codes . . . . . . . . . . . . . . . . . . . . 11
4 Rank Analysis for QC-LDPC Codes 15
4.1 Rank Analysis of Parity-Check Matrices with One Row-Block . . . . . . . . 15
4.2 Rank Analysis of Parity-Check Matrices with Two Row-Blocks . . . . . . . . 18
4.3 Rank Analysis of Parity-Check Matrices with Three Row-Blocks . . . . . . . 20
4.4 Rank Computing Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 Conclusion 31

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