以馬古勒斯碼與拉瑪紐俊-馬古勒斯碼為基礎所建構的低密度同位元檢查碼

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研究生：	張孟華 Meng-Hua Chang
論文名稱：	以馬古勒斯碼與拉瑪紐俊-馬古勒斯碼為基礎所建構的低密度同位元檢查碼 Construction of LDPC Codes from Margulis Codes and Ramanujan-Margulis Codes
指導教授：	呂忠津 Chung-Chin Lu
口試委員:
學位類別：	碩士 Master
系所名稱：	電機資訊學院 - 電機工程學系 Department of Electrical Engineering
論文出版年：	2004
畢業學年度：	92
語文別：	英文
論文頁數：	65
中文關鍵詞：	低密度同位元檢查碼、馬古勒斯碼、拉瑪紐俊-馬古勒斯碼、蓋勒格、代數方法建構
外文關鍵詞：	LDPC codes, Margulis codes, Ramanujan-Margulis codes, Gallager, algebraic construction
相關次數：	點閱：3 下載：0
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早在六零年代，蓋勒格(Gallager)即在他的論文中提出一種錯誤更正碼，稱為低密度同位元檢查碼(LDPC Codes)。但在接下來的三十年，它被大多數的人遺忘。直到1993年渦輪碼(Turbo Codes)興起後，才開始有更多人對它做詳細的探討與研究，目前它已成為最熱門的錯誤更正碼之一。與已往的錯誤更正碼最大不同的地方在於，之前的碼是著眼在尋找擁有高度組織化及較大的最小距離(minimum distance)，相對的它的解碼處會較為簡單。但實際上，隨機尋找的碼會有很高的機率是足夠好的，因此，在蓋勒格的論文中，低密度同位元碼的建構方法是使用隨機方式，只是它的解碼處是使用循環解碼，利用這種方式我們可以得到很好的錯誤更正效果。

在這篇論文中，我們研究和分析兩種用代數方法所建構出來的低密度同位元檢查碼的效能。它們分別是馬古勒斯碼(Margulis Codes)以及拉瑪紐俊-馬古勒斯碼(Ramanujan-Margulis Codes)。藉著使用隨機尋找的方法，我們可以建構出一種馬古勒斯碼，它相對應的圖形中所擁有最短循環路徑的長度在某些情況下將較原本馬古勒斯論文中所建構的馬古利斯碼長。當傳輸的碼短時，我們建構出的馬古勒斯碼相對之下將擁有更佳的錯誤更正效果。此外，我們還提出藉由以上兩種用代數方法所建造的碼來建構不規則的低密度同位元檢查碼。它們比使用隨機建構的不規則碼有更佳的錯誤更正效果。

In this thesis, we investigate and analyze the performance of two algebraic constructions
of low-density parity-check codes. They are Margulis codes [18] and Ramanujan-Margulis
codes [19], [9], [29]. By a random search method, we obtain Margulis codes with larger girths.
We also describe a method to construct irregular codes based on these two algebraically
constructed codes. They have better performance than randomly constructed codes. They
also outperform PEG codes in low SNR region.

Low-Density Parity-Check Codes 1
1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Definitions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4 Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.2 Gallager's hard-decision decoding algorithm [7] . . . . . . . . . . . . 7
4.3 Belief-propagation decoding algorithm: . . . . . . . . . . . . . . . . . 10
Margulis Codes and Ramanujan-Margulis Codes Constructions 11
1 Margulis codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1 Cayley Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Application to LDPC codes . . . . . . . . . . . . . . . . . . . . . . . 13
2 Ramanujan-Margulis codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1 Ramanujan Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Expander . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Explicit Construction of Ramanujan Graphs . . . . . . . . . . . . . . 17
2.4 An application to LDPC codes . . . . . . . . . . . . . . . . . . . . . 20
Other LDPC Code Constructions 22
1 Regular LDPC Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.1 Gallager's Codes [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2 Mackay-1A [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3 Progressive Edge-Growth Tanner Graphs [8] . . . . . . . . . . . . . . 23
2 Irregular LDPC Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 Mackay-2A [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Luby et. al.'s Method [13] . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Richardson et. al.'s Method [26] . . . . . . . . . . . . . . . . . . . . . 27
2.4 Random Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Heuristic Search Method [20] . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Progressive Edge-Growth Tanner Graphs [8] . . . . . . . . . . . . . . 29
2.7 PEG-Random Method . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Properties of LDPC Codes 30
1 Upper Bound for the Girth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Lower Bound for the Minimum Distance of LDPC Codes . . . . . . . . . . . 32
3 Concentration and Convergence to the Cycle-Free Case . . . . . . . . . . . . 33
Modification to Margulis and Ramanujan-Margulis Codes 35
1 Extended Regular Margulis Codes . . . . . . . . . . . . . . . . . . . . . . . . 35
2 Irregular Margulis Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.1 Irregular Margulis-A . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Irregular Margulis-B . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Irregular Mar-Random method . . . . . . . . . . . . . . . . . . . . . 37
3 Irregular Ramanujan-Margulis Codes . . . . . . . . . . . . . . . . . . . . . . 37
3.1 Irregular Raman-Random method . . . . . . . . . . . . . . . . . . . . 37
Simulation 38
1 Margulis Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.1 q=5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.2 q=7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.3 q=11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.4 q=13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2 Ramanujan-Margulis code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.1 p=5, q=13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2 p=5, q=17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Conclusion

                                

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