ALGEBRAIC CODING THEORY, COMBINATORIAL DESIGNS AND ASSOCIATION SCHEMES

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。

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.