距离正则图的Terwilliger代数及其在码中的应用

Terwilliger代数理论是代数组合论的重要研究内容，它是研究Q-多项式距离正则图分类的有效方法，而且与编码理论有着紧密的联系。本项目研究关于距离正则图的Terwilliger代数的结构及其应用问题，包括以下两个方面：1. 刻画折叠Johnson图、n-立方图的半图、折叠n-立方图的半图等一些薄Q-多项式距离正则图的Terwilliger代数结构；2. 利用Terwilliger代数的准对角化形式和半定规划算法确定上面这些图中码的上界。本项目的研究不仅可以进一步丰富Terwilliger代数理论，也将对距离正则图及其相关学科的发展起到积极的推动作用。

The Terwilliger algebra theory is an important object in algebraic combinatorics, which is a useful tool in studying the classification of Q-polynomial distance-regular graphs and also closely related to the coding theory. This project will investigate the Terwilliger algebra of distance-regular graphs. It includes the following two aspects: 1. We describe the Terwilliger algebras for some thin Q-polynomial distance-regular graphs such as the folded Johnson graph, the halved n-cube and the halved folded n-cube；2. We study codes upper bounds for the above graphs by semidefinite programming involving the block-diagonalization of the Terwilliger algebra. The important features of this research are that it will not only enrich the Terwilliger algebra theory further, but also contribute well for the developments of distance-regular graphs and related areas.