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.

应用Jacobi和及相关数论结果，证明了由AG(2n，3)中的2-平坦构成的2-设计比已知的分解可以分解成更多的子设计。同时，通过考察割圆陪集中点的分布，给出了得到的子设计个数的精确估计。本课题的初衷是寻找可应用于最佳光正交码的Steiner四元系的构造，但其结果最终被证明适用于秘密共享方案和量子跳跃编码。对于冲突避免码，通过引入扩展奇序列的新概念，给出了长度为n≡4(Mod 8)、重量为3的最优码的直接构造。结果，利用先前已知的结果，长度和重量为偶数的最佳冲突避免码的大小的谱完全确定。

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

Optimal equi-difference conflict-avoiding codes of length n=2^a3^bm and weight four

长度n=2^a3^bm、权重4的最优等差冲突避免码

• 发表时间: 2013
• 作者: Y. Lin;M. Mishima;M. Jimbo
• 通讯作者: M. Jimbo

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

Equi-difference conflict-avoiding codes of length n = 2^a3^bm and weight four

长度为 n = 2^a3^bm 且权重为 4 的等差冲突避免码

• 发表时间: 2013
• 作者: Y. Lin;M. Mishima;M. Jimbo
• 通讯作者: M. Jimbo

Tight Equi-Difference Conflict-Avoiding Codes of Odd Length and Weight Three

奇长奇重三紧等差冲突避免码

• 发表时间: 2012
• 作者: Y. Lin;M. Mishima;J. Sato;M. Jimbo
• 通讯作者: M. Jimbo