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研究生： 劉豐源
Liu, Feng-Yuan
論文名稱： 使用半正定規劃解決球面三距離集合問題
Semidefinite Programming Bounds For Spherical Three-Distance Sets
指導教授： 李哲榮
Lee, Che-Rung
俞韋亘
Yu, Wei-Hsuan
口試委員: 傅恆霖
Fu, Hung-Lin
林延輯
Lin, Yen-Chi
學位類別： 碩士
Master
系所名稱： 電機資訊學院 - 資訊工程學系
Computer Science
論文出版年： 2019
畢業學年度： 107
語文別： 英文
論文頁數： 37
中文關鍵詞： 球面多距離集合 、球面編碼 、球面設計 、半正定規劃 、凸優化
外文關鍵詞： spherical few distance set, spherical codes, spherical design, semidefinite programming, convex optimization
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    • 球面三距離集合(spherical three-distance set)是指一個有限的單位向量集合，使得該集合滿足兩兩相異向量間的距離只有三種(也就是只有三種內積值)。最大基數的球面三距離集合是球面編碼理論裡的一個經典問題。
    在此本篇論文，我們使用了半正定規劃方法，在許多維度中得到了更緊的上界。在7維，我們將上界從91降低到84；在23維，我們把上界從2301降低到2300，並且證明2300個點是最大基數的球面三距離集合在23維中。

    A spherical three-distance set is a fi nite collection X of unit vectors in R^n such that for each pair of distinct vectors has three inner product values. The maximum cardinality of spherical three-distance set is a classic problem in spherical coding
    theorem.
    In this thesis, we use the semidefi nite programming method to improve the upper bounds of spherical three-distance sets for several dimensions. We improve the upper bounds for spherical three-distance sets in R^7 from 91 to 84 and we prove that maximum size of spherical three-distance sets is 2300 in R^23.

    1 Introduction 1 2 Previous method 4 3 Semide nite programming method 8 4 Discrete sampling points with Nozaki theorem 11 5 Continuous interval with sum of squares method 14 6 Experiments 19 7 Discussion and conclusion 29 Bibliography 31 A Uniform distribution on spherical space 34 B Correction on related paper 36

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