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研究生： 曾秀華
Tseng, Hsiu-Hua
論文名稱： 利用一個生成元的偽循環碼探討(n,1,ν)卷積碼
Explore (n,1,ν) Convolutional Codes by 1-Generator Quasi-cyclic Codes
指導教授： 柯文峰
Ke, Wen-Fong
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年： 2017
畢業學年度： 105
語文別： 英文
論文頁數： 42
中文關鍵詞： 卷積碼 、里德-所羅門碼 、最小自由距離
外文關鍵詞： convolutional codes, Reed-Solomon codes, minimum free distance.
相關次數： 點閱：145  下載：9 
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    • 利用準循環碼 (quasi-cyclic codes)和卷積碼 (convolutional codes)能具有相同最小(自由)距離，與里德-所羅門碼 (Reed-Solomon codes)可以設計最小距離的特性，構造出不易發生解碼錯誤的卷積碼 (convolutional codes)。

    According to that quasi-cyclic codes and convolutional codes can have the same minimum (free) distance, and that Reed-Solomon codes can be designed minimum distance, we can construct convolutional codes with lower error probability of decoding.

    Contents 1 Introduction 3 1.1 Background of coding theory . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The difference codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Research motives and purposes . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Background 4 2.1 Convolutional codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Viterbi decoding algorithm . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.2 Decoding error probability on a binary symmetric channel (BSC). . 15 2.2 Block codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1 Cyclic codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1.1 BCH codes . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1.2 Reed-Solomon codes . . . . . . . . . . . . . . . . . . . . . 19 2.2.2 Quasi-cyclic codes (QC codes) . . . . . . . . . . . . . . . . . . . . . 21 3 A link between convolutional codes and quasi-cyclic codes 24 4 Construct 1-generator quasi-cyclic codes 26 5 A lower bound of the minimum distances of quasi-cyclic codes which are associated with Reed-Solomon codes 37 6 Conclusion 39 7 Appendix 41 7.1 GAP : find wight enumerations of quasi-cyclic codes . . . . . . . . . . . . . 41 7.1.1 Generator polynomial ˜ g(x) = (x 14 + x 13 + x 11 + x 7 + 1,0,0). . . . . 41 7.1.2 Generator polynomial ˜ g(x) = (x 14 +x 13 +x 12 +x 11 +x 10 +x 8 +x 6 +x 5 + x 4 +x 2 ,x 13 +x 12 +x 10 +x 4 +x 3 +x+1,x 13 +x 12 +x 10 +x 5 +x 4 +x 3 +x 2 ). 41 7.2 The cyclotomic cosets modulo 7, 73 and 273 . . . . . . . . . . . . . . . . . 42

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