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研究生: 賴虹汶
Lai, Hung-Wen
論文名稱: 四元代數碼及網格的關係
A Survey on The Relations between Quaternary Codes and Lattices
指導教授: 林正洪
Lam, Ching-Hung
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: 2008
畢業學年度: 96
語文別: 英文
論文頁數: 52
外文關鍵詞: the Leech lattice, self-dual, Niemeier lattice, quaternary code
相關次數: 點閱:69下載:4
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  • In this thesis, we shall give a survey on the relations between lattices and quaternary codes. In particular, we will discuss the construction of Niemeier lattices from self-dual quaternary codes. We introduce some definitions and properties about quaternary codes in the beginning of the thesis. Then we study the relation of lattices and quaternary codes by Construction A. Moreover, we use the method of Kitazume and Harada to embed all Niemeier lattices into the Leech lattice.

    1 Introduction 2 2 Introduction for Quaternary Codes 5 2.1 Basic Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Quaternary Linear Codes and Their Generator Matrices . . . . . . 7 3 Weight Enumerators and Distance Enumerators 9 3.1 Weight Enumerators of Quaternary Codes . . . . . . . . . . . . . 9 3.2 Krawtchouk Polynomial . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Distance Enumerators of Binary Codes . . . . . . . . . . . . . . . 20 4 The Relations between Quaternary Codes and Binary Codes 26 4.1 The Gray Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 Binary Images of Quaternary Codes . . . . . . . . . . . . . . . . 29 4.3 Linearity Condition . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.4 Binary Codes Associated with a Quaternary Linear Code . . . . . 34 5 Z4-Linearity and Z4-Nonlinearity of RM(r, m) 36 5.1 The Z4-Linearity of Some RM(r, m) . . . . . . . . . . . . . . . . 36 5.2 The Z4-Nonlinearity of Some RM(r, m) . . . . . . . . . . . . . . 37 6 Lattice 39 6.1 Introduction for Lattice . . . . . . . . . . . . . . . . . . . . . . . 39 6.2 Construction A . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.3 Quaternary Code Constructions for the Niemeier Lattices . . . . . 45 Reference 51

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