Constructions of optimal optical orthogonal codes and conflict-avoiding codes derived from combinatorial designs

组合设计的最优光学正交码和冲突避免码的构造

• 批准号: 22540121
• 负责人: MISHIMA Miwako
• 金额: $ 2.33万
• 依托单位: Gifu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2010
• 资助国家: 日本
• 起止时间: 2010 至 2012
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-22540121/
• 关键词: 組合せ論; Affine geometry; decomposition; 2-design; Jacobi sum; Weil sum; conflict-avoiding code; multiplicative order; cyclotomic polynomial; Skolem type sequence; Conflict-avoiding code; Design decomposition; 2-flats in an affine geometry

By applying Jacobi sums and some related number theoretic results, it is shown that the 2-design formed by the 2-flats in AG(2n,3) can be decomposed into more subdesigns than a previously known decomposition. At the same time, exact evaluation of the number of the resulting subdesigns is also demonstrated by examining the distribution of points in cyclotomic cosets. The original purpose of this theme was to find constructions of Steiner quadruple systems which can be applied to optimal optical orthogonal codes, but the result eventually turned out to be applicable to secret sharing scheme and quantum jump codes. As for conflict-avoiding codes, direct constructions for optimal codes of length n≡4 (mod 8) and weight 3 are provided by bringing in a new concept called an extended odd sequence. As a consequence, with previously known results, the spectrum of the size of optimal conflict-avoiding codes of even length and weight 3 is completely settled.

通过应用Jacobi总和和一些相关的数字理论结果，可以表明，与先前已知的分解相比，Ag（2n，3）中2-Flats形成的2个设计可以分解为更多的细胞设计。同时，还通过检查环形磁盘中点的分布来证明对所得子设计的数量的精确评估。该主题的最初目的是找到可以应用于最佳光学正交代码的Steiner四倍系统的构造，但是结果最终证明适用于秘密共享方案和量子跳跃代码。至于避免冲突的代码，通过引入一个称为扩展奇数序列的新概念来提供长度n程（mod 8）和重量3的最佳代码的直接构造。结果，有了先前已知的结果，完全解决了避免长度和重量3的最佳避免冲突代码的频谱。

项目成果

Optimal Conflict-Avoiding Codes of Even Length and Weight 3

• DOI: 10.1109/tit.2010.2069270
• 发表时间: 2010-11
• 期刊: IEEE Transactions on Information Theory
• 影响因子: 2.5
• 作者: Hung-Lin Fu;Yi-Hean Lin;Miwako Mishima

Mutually orthogonal t-designs over C related to quantum jump codes

与量子跳跃码相关的 C 上的相互正交 t 设计

• 发表时间: 2012
• 作者: Ichikawa;H;Otsuka;Y.;Kanazawa;S.;Yamaguchi M. K.;& Kakigi;R.;Masakazu Jimbo
• 通讯作者: Masakazu Jimbo

Quantum jump codes and mutually orthogonal partially balanced t-designs over C

C 上的量子跳跃码和相互正交部分平衡 t 设计

• 发表时间: 2011
• 作者: G.Ishikawa;Y.Machida;M.Takahashi;Toshiyuki Kobayashi;M.Jimbo
• 通讯作者: M.Jimbo

アフィン幾何からできるデザインの分解について

关于仿射几何设计的分解

• 发表时间: 2010
• 作者: 長谷川潤;籾原幸二;三嶋美和子;神保雅一
• 通讯作者: 神保雅一

Recursive constructions of t-SEEDs related to quantum jump codes

与量子跳跃码相关的 t-SEED 的递归构造

• 发表时间: 2010
• 作者: 林怡伶;神保雅一
• 通讯作者: 神保雅一