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研究生:游曜禧
研究生(外文):Yau-Shi You
論文名稱:編碼、網格及其應用
論文名稱(外文):A Survey on Codes, Lattices and their application
指導教授:林正洪林正洪引用關係
指導教授(外文):Ching-Hung Lam
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:2003
畢業學年度:91
語文別:英文
論文頁數:29
外文關鍵詞:codelatticeLeech lattice
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  Codes and lattices are very important subject in combinatorics. They have many applications in telecommunication, design theory, finite group theory as well as many different fields in electronic engineering and physics.

  In this thesis, we will give a survey on the properties of certain codes and lattices. In particular, we will concentrate on the construction of certain unimodular lattice by using linear codes. We will also discuss their application to other fields such as Lie algebra and finite group.
1 Introduction                 2

2 Code                   4
 2.1 Basic De nitions. . . . . . . . . . . . . . . . . . . . . . . . . . .4
 2.2 Certain important codes . . . . . . . . . . . . . . . . . . . . .6
  2.2.1 Hamming code . . . . . . . . . . . . . . . . . . . . . . . .6
  2.2.2 Golay code. . . . . . . . . . . . . . . . . . . . . . . . . . .7

3 Lattices                   9
 3.1 The lattices E6, E7 and E8 . . . . . . . . . . . . . . . . . . .12
 3.2 Unimodular lattices and Leech lattice . . . . . . . . . . . 15

4 Construction of Lattices from Codes        18

5 Gluing theory                22
 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
 5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
 5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27

Bibliography                 28
[1] R. Bacher and B.B. Venkov, Rseaux entiers unimodulaires sans racines en di-mension 27 et 28, Rseaux euclidiens, designs spheriques et formes modulaires, 212-267, Monoger. Enseign. Math, 37. Enseignement Math, Geneva 2001.

[2] A. Bonnecaze, P. Sole and A. R. Calderbank, Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory 41, 366-377, 1995.

[3] W. Bosma and J. Cannon, Handbook of Magma Functions, School of Mathematics and Statistics, University of Sydney, Sydney, July 22, 1999.

[4] J. H. Conway and N. J. A. Sloane, On the enumeration of lattices of determinant one, J. Number Theory 15, 83-94, 1985.

[5] J. H. Conway and N. J. A. Sloane, Sphere Packing, Lattices and Groups, 2nd ed., Springer-Verlag, New York, 1993.

[6] S. T. Dougherty, M. Harada and P. Sole, Shadow codes over Z4, Finite Fileds and Their Appl. 7, 507-529, 2001.

[7] I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Academic-Press, Inc, London, 1988.

[8] T. A. Gulliver and M. Harada, Orthogonal Frames in the Leech Lattice and a Type II Code over Z22, Journal of Combinatorial Theory, Ser. A 95, 185-188, 2001.

[9] M. Harada, P. Sol e, and P. Gaborit, Self-Dual Codes over Z4 and Unimodular Lattices: A Survey, Algebras and Combinatiorics (Hong Kong 1997), 255-275, Spring, Singapore, 1999.

[10] R. Hill, A First Course in Codeing Theory, Clarendon-Press, Oxford, 1986.

[11] M. Kervaire, Unimodular lattices with a complete root system, Ens. Math. 40, 59-104, 1994.

[12] M. Kitazume, C.-H. Lam, and H. Yamada, A class of vertex operator algebras constructed from Z8 codes, J. Algebra 242, 338-359, 2001.

[13] M. Kitazume, C.-H. Lam, and H. Yamada, Decomposition of the Moonshine Vertex Operator Algebra as Virasoro Modules, J. Algebra 226, 893-919, 2000.

[14] H. Koch and B. B. Venkov, Ueber ganzahlige unimodulare Gitter, J. reine angew. Math. 398,144-168, 1989.

[15] J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer Verlag, New York, Heidelberg, Berlin, 1972.

[16] C.-H. Lam, Fusion rules for the Hamming code vertex operator algebra, Comm. in Algebra 29(5), 2125-2145, 2001.

[17] C.-H. Lam and H. Yamada, Z2 Z2 codes and vertex operator algebra, J. Algebra 224, 268-291, 2000.

[18] R. V. Moody and A. Pianzola, Lie Algebras With Triangular Decompositions, John-Wiley-Sons, New York, 1995.

[19] H. V. Niemeier, De nete quadratische Formen der Dimension 24 und Diskrim-inante 1, J. Number Theory 5, 142-178, 1973.

[20] H. G. Quebbemann, Zur Klassi cation unimodularer Gitter mit Isometrie von Primzahlordnung, J. reine angew. Math. 326, 158-170, 1981.
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