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研究生:顏銘村
研究生(外文):Ming-Cun Yan
論文名稱:使用伽諾瓦場提出循環置換碼
論文名稱(外文):Construction of Cyclically Permutable Codes Using Finite Field Fourier Transform
指導教授:陳後守
指導教授(外文):Hou-Shou Chen
口試委員:黃育銘楊谷章
口試委員(外文):Yuh-Ming HuangGuu-Chang Yang
口試日期:2017-07-26
學位類別:碩士
校院名稱:國立中興大學
系所名稱:電機工程學系所
學門:工程學門
學類:電資工程學類
論文種類:學術論文
論文出版年:2017
畢業學年度:105
語文別:中文
論文頁數:43
中文關鍵詞:頻域循環碼循環置換碼
外文關鍵詞:Cyclic codeEncoding
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本論文將探討如何從循環碼(cyclic code)中建構循環置換碼(cyclically permutable code, CPC)。近年來循環置換碼越來越重要,它可應用於通信網絡,光纖通訊與影像處理中,例如多重存取碰撞於無反饋渠道,跳頻擴頻通信通道,光正交碼與數位浮水印等。循環碼是區塊碼且滿足任何碼字(codeword)經由循環位移(cyclic shift)所產生的碼字仍然是此循環碼的碼字。而長度 的循環置換碼的定義就是那些擁有循環階數(cyclic order)為 的字碼且碼字循環位移後所產生的這些 個碼字只能在循環置換碼中出現一次。本論文之精神將探討如何透過循環碼之特性與有限場傅立葉轉換(Galois field Fourier transform, GFFT),從循環碼中有效地建構出循環置換碼。我們提出了使用伽諾瓦場傅立葉轉換的兩個概念:共軛特性和位移特性,使用頻域編碼的方法來建構循環置換碼。我們已經分析從(7,4,3)與(17,9,5)循環碼去建構循環置換碼,藉由此將建構出任何長度為質數之循環置換碼。也將以(15,11,3)與(21,12,5)循環碼建構出循環置換碼為例,建構出任何長度為合成數之循環置換碼。我們的目標是希望在循環碼中,利用頻域編碼或時域編碼中找出一套有效且快速的方法來建構循環置換碼並且將此碼應用於通信網絡,光纖通訊與影像處理中。
In this study, we investigates the construction of cyclically permutable code (CPC) from a cyclic code. In recent years, the cyclically permutable codes have become increasingly important and have been applied in the communication network, optical communication, and image processing, such as multiple access collision channels without feedback, frequency-hopping spread spectrum communication channels, optical orthogonal codes, and digital watermarking. A cyclic code C is a linear block code such that any cyclic shift of a codeword in C is another codeword in C. A CPC of length n is a block code such that each codeword has full cyclic order n and all codewords are cyclically distinct.
The main theme of this stydy is to construct a CPC from a cyclic code using the Galois field Fourier transform (GFFT).we used the conjugate property and translation property of GFFT, and then use the frequency-domain encoding of a cyclic code to find a CPC. In this study, the results of the construction of a CPC from the cyclic codes (7,4,3) and (17,9,5) will be used to show how to construct a CPC with prime length. Similarly, the results of the construction of a CPC from the cyclic codes (15,11,3) and (21,12,5) will be used to show how to construct a CPC with non-prime length. Our main goal of this study is to find a systematic and efficient algorithm to find a CPC from a cyclic code, and then apply these CPCs to the communication network, optical communication, and image processing.
目錄
摘要 ii
Abstract iii
圖目錄 vi
Chapter 1 前言 1
Chapter 2 簡介 3
2.1 線性區塊碼 4
2.1.1線性區塊碼介紹和圖解 4
2.2 漢明距離及漢明權重 7
2.2.1最小漢明距離 7
2.2.2 漢明權重 8
2.2.3 錯誤更正與偵錯能力 10
2.3循環碼與循環置換碼 12
2.3.1循環碼 12
2.3.2循環置換碼 13
Chapter 3 伽諾瓦場傅立葉轉換 15
3.1伽諾瓦場 15
3.1.1 質數伽諾瓦場 16
3.1.2 擴張伽諾瓦場 17
3.2 共軛組與循環碼造碼 18
3.2.1 共軛組 18
3.2.2 藉由共軛組建構循環碼 21
3.3伽諾瓦場傅立葉轉換 23
Chapter 4 於循環碼中提出循環置換碼 26
4.1利用伽諾瓦場傅立葉轉換造循環碼 27
4.2 使用伽諾瓦場傅立葉轉換於循環碼中提出循環置換碼 29
4.2.1方法一、當循環碼長n為質數且為原始長度 29
4.2.2方法二、當循環碼長n為質數且為非原始長度 31
4.2.3方法三、當循環碼長n為非質數且為原始長度 34
4.2.4方法四、當循環碼長n為非質數且為非原始長度 37
Chapter 5 結論 40
參考文獻 41
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