ALGEBRAIC CODING THEORY, COMBINATORIAL DESIGNS AND ASSOCIATION SCHEMES

推广了特征为2的任意有限域上的II码的概念，得到了这类码的质量公式。这使得我们能够对几乎所有二进制长度最多为32的II型码进行分类，我们的结果在澳大利亚举行的第14届AAECC研讨会上公布。我们解决了一个长期存在的关于紧球面设计的问题，这一结果在2002年在荷兰举行的国际组合学会议上公布。我们的结果使用了模形式理论，我们参加了2004年的国际模形式会议，以寻找组合学和模形式理论之间的更多联系。我们还探讨了具有二进制码的整格和具有普通组合设计的球面设计之间的联系。我们证明了一类与二元自对偶码相关的自正交5-设计的唯一性，并提交了一篇关于这一结果的论文。

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.