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研究生:王勝勇
研究生(外文):Shen-Yong Wang
論文名稱:佈於整數模六十四的二次剩餘碼
論文名稱(外文):Quadratic Residue Codes over Z64
指導教授:俞勇俞勇引用關係
指導教授(外文):Yung Yu
學位類別:碩士
校院名稱:國立成功大學
系所名稱:數學系應用數學碩博士班
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2006
畢業學年度:94
語文別:英文
論文頁數:111
中文關鍵詞:二次剩餘碼
外文關鍵詞:idempotent generatorquadratic residue codes
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二次剩餘碼是一種特殊的循環碼, 首先定義出它的人是Andrew Gleason, 對於編碼的長度而言, 許多適當的二次剩餘碼有著極高的(high)最小非零量(minimum weights), 這也使得這類的碼前景看好, 佈於整數模 的二次剩餘碼在Bonnecaze, Solé 以及 Calderbank [Bo-So-Ca] 和 Pless 以及 Qian [Pl-Qi] 等人的研究之下, 發現出許多漂亮的結果, 在[Chiu-Yu-Yau]一文中, 作者對於整數模八的二次剩餘碼做了一些詳細的研究, 而在這篇論文中, 吾人利用類似[Chiu-Yu-Yau]的手法, 找出了整數模六十四的二次剩餘碼的生成元(idempotent), 進而生成出佈於整數模六十四的二次剩餘碼, 並證明出這些碼類似於佈於體上的碼, 都有著許多好的特性, 在[Ka]一文中, Kanwar已對於整數模q^m且長度為p的二次剩餘碼得出了一般性的結果, 此處的p是一不為2的質數並在模p下為+1或-1, 而Kanwar也引進了擴充二次剩餘碼的觀念並找出了它的對偶碼, 本篇論文的另一目標是證明出整數模六十四的擴充二次剩餘碼有著大量的同構群, 藉由[Ma-Sl]中所用的有效力的排列解碼方法, 使得這些同構群對於解出這些二次剩餘碼有著極高的效力.
A set of n tuples over Z64 is called a code over Z64 or a Z64 code if it is a Z64 module. A particularly interesting
family of cyclic codes is the quadratic residue codes. Quadratic residue codes were first defined by Andrew Gleason. The minimum weights of many modest quadratic codes are quite high for the codes lengthes, making this class of codes promising. The Zq^m- quadratic residue codes were studied by beautiful works of Bonnecaze, Solé and Calderbank [Bo-So-Ca] and Pless and Qian [Pl-Qi] . In [Chiu-Yu-Yau] , the authors studied the Z8 quadratic residue codes in some detail. In this paper, we define Z64 quadratic residue codes in terms of their idempotent generators and show that these codes also have many good properties which are analogous in many respects to the properties of quadratic residue codes over a field. In [Ka], Kanwar has general results on quadratic residue Zq^m-codes of length p where p is an odd prime congruent to 1 or -1 modulo 4q . The concepts of extended quadratic residue Zq^m-codes is introduced in [Ka] and their duals are obtained. The purpose of this paper is to show that extended quadratic residue codes over Z64 have large automorphism groups which will be useful in decoding these codes by using the powerful permutation decoding methods described in [Ma-Sl].
Contents

CHAPTER 1  Introduction P1
   
CHAPTER 2  Quadratic Residue Codes over  P26
2.1  Idempotent generator of QR codes  over  P26
2.2  Properties of QR codes over  P51
2.3  Automorphism group of the  extended QR code over  P74
Reference   P110
[Bo-So-Ca]:A. Bonnecaze, P. Solé and A.R. Calderbank,Quaternary
Quadratic Residue Codes and Unimodular Lattices, IEEE Trans. Inform. Theory, 41(1995), 366-377.
[Di]:L.E. Dickson, Linear Groups, Dover Publications, Inc. NY, 1958.
[Ka-Lp]:P. Kanwar and S. López-Permouth, Cyclic codes over the integers modulo q^m , Finite Fields and Their Applications, 3(1997), 334-352.
[Ka]:P. Kanwar, Quadratic Residue Code over Integer Module q^m, Contemporary Mathematics, 259(2000), 299-312.
[Le-Ma-Pl]:J.S. Leon, J.M. Masley, and V. Pless, Duadic Codes, IEEE Trans. Inform. Theory, 30(1994), 709-714.
[Ma-Sl]:F.J. MacWilliams and N.J.A. Sloane, Theory of Error-Correcting Codes, North Holland, Amsterdam, 1978.
[Pl]:V. Pless, Introduction to the Theory of Error-Correcting Codes, Second Edition, Wiley Interscience, 1989.
[Pl-Qi]:V. Pless and Z. Qian, Cyclic Codes and Quadratic Residue Codes over Z4 , IEEE Transactions on Information Theory, 42, No.5(1996), 1594-1600.
[Pe]:O. Perron, Bemerkungen über die Verteilung der quadratischen Reste, Math. Zeit., 56(1952), 122-130.
[Chiu-Yu-Yau]:
Mei Hui Chiu, Yung Yu, Stephen S.-T. Yau,  Cyclic Codes and Quadratic Residue Codes, Advances in Applied Mathematics 25, 12-33(2000).
[許琮琳]:佈於整數模拾陸的二次剩餘碼,國立成功大學應數所碩士論文, June 1999.
[郭威良]:佈於整數模三十二的二次剩餘碼,國立成功大學應數所碩士論文, June 2003.
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