ALGEBRAIC CODING THEORY, COMBINATORIAL DESIGNS AND ASSOCIATION SCHEMES

代数编码理论、组合设计和关联方案

• 批准号: 12440030
• 负责人: MUNEMASA Akihiro
• 金额: $ 3.9万
• 依托单位: TOHOKU UNIVERSITY (2003)Kyushu University (2000-2002)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440030/
• 关键词: ALGEBRAIC COMBINATORICS; SELF-DUAL CODE; DISTANCE-REGULAR GRAPH; FINITE FIELD; SPIN MODEL; MODULAR FORM; LATTICE; 代数的符号理論; モジュラー形成

A generalization of the concept of a type II code has been obtained for arbitrary finite fields of characteristic two, and the mass formulas for such codes have been obtained. This allowed us to classify nearly all type II codes whose binary lengths are at most 32, and our results were presented in the 14th AAECC symposium in Australia. We have solved a long standing problem of the existence of tight spherical designs, and this result was presented in an international conference on combinatorics in the Netherlands in 2002.Our result uses the theory of modular forms, and we participated in an international conference on modular forms in 2004 in order to find more connections between combinatorics and the theory of modular forms. We also explored the connections between integral lattices with binary codes, and spherical designs with ordinary combinatorial designs. We have established a uniqueness of a certain self-orthogonal 5-design related to a binary self-dual code, and we have submitted a paper about this result.

对于特征二的任意有限域，得到了II型码概念的推广，并得到了此类码的质量公式。这使得我们能够对几乎所有二进制长度最多为 32 的 II 类代码进行分类，我们的结果已在澳大利亚第 14 届 AAECC 研讨会上发表。我们解决了一个长期存在的紧球设计存在性问题，这个结果在2002年荷兰的国际组合学会议上发表。我们的结果使用了模形式的理论，我们参加了国际模联会议2004 年，为了找到组合数学和模形式理论之间的更多联系。我们还探索了积分格与二进制代码之间的联系，以及球形设计与普通组合设计之间的联系。我们已经建立了与二进制自对偶码相关的某种自正交5设计的唯一性，并且我们已经提交了关于该结果的论文。

项目成果

M.Klin, A.Munemasa, M.Muzychuk, P.-H.Zieschang: "Directed strongly regular graphs via coherent algebras"Linear Algebra and Applications. 377. 83-109 (2004)

M.Klin、A.Munemasa、M.Muzychuk、P.-H.Zieschang：“通过相干代数有向强正则图”线性代数和应用。

T.Aaron Gulliver and Masaaki Harada: "Double circulant self-dual codes over GF (5)"Ars Combinatoria. 56. 3-13 (2000)

T.Aaron Gulliver 和 Masaaki Harada：“GF 上的双循环自对偶代码 (5)”Ars Combinatoria。

M.Harada, A.Munemasa, K.Tanabe: "Extremal self-dual [40,20,8] codes with covering radius 7"Finite Fields and Their Applications. 10. 183-197 (2004)

M.Harada、A.Munemasa、K.Tanabe：“覆盖半径为 7 的极值自对偶 [40,20,8] 码”有限域及其应用。

Akira Hiraki,Kazumasa Nomura and Hiroshi Suzuki: "Distance-regular graphs of valency 6 and α_1=1"Journal of Algebraic Combinatorics. 11. 101-134 (2000)

Akira Hiraki、Kazumasa Nomura 和 Hiroshi Suzuki：“价数 6 和 α_1=1 的距离正则图”代数组合学杂志 11. 101-134 (2000)。

T.Aaron Gulliver, Masaaki Harada: "Optimal Double Circulant Z_4-Codes"Lecture Notes in Computer Science. 2227. 122-128 (2001)

T.Aaron Gulliver、Masaaki Harada：“最优双循环 Z_4 代码”计算机科学讲义。