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.

对特征为2的任意有限域，推广了II型码的概念，得到了II型码的质量公式。这使我们能够对二进制长度最多为32的几乎所有II型代码进行分类，我们的结果在澳大利亚举行的第14届AAECC研讨会上发表。我们解决了一个长期存在的紧密球形设计的问题，这个结果在2002年荷兰的一个国际组合学会议上发表。我们的结果使用了模形式理论，我们参加了2004年关于模形式的国际会议，以便在组合学和模形式理论之间找到更多的联系。我们还探讨了积分格与二进制编码之间的联系，以及球面设计与普通组合设计之间的联系。我们建立了与二进制自对偶码相关的某自正交5-设计的唯一性，并就此结果提交了一篇论文。

项目成果

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。

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。

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)。

E.Bannai, M.Harada, T.Ibukiyama, A.Munemasa, M.Oura: "Type II codes over F_2+uF_2 and applications to Hermitian modular forms"Abhandlungen Mathematischen Seminar der Universitat Hamburg. 73. 13-42 (2003)

E.Bannai、M.Harada、T.Ibukiyama、A.Munemasa、M.Oura：“F_2 uF_2 上的 II 型代码及其在 Hermitian 模形式中的应用”汉堡大学 Abhandlungen 数学研讨会。