The Terwilliger Algebras Of Some Distance-Transitive Graphs And Their Applications On The Codes

The Terwilliger algebra of distance-regular graph is an important topic in algebraic combinatorics theory,which is not only a useful tool in studying the classification of Qpolynomial distance-regular graphs,but also tightly relat.ed to coding t heory,design theory,Lie algebra,quantum algebra and so on.The Terwilliger algebra of distance-transitive graph,as one subclass of distance-regular graph,has not been researched uniformly.In this thesis,we first investigate the Terwilliger algebra of one class of distance-transitive graph satisfying a natural assumption,then we concretely give the Terwilliger algebras of bipartite half of n-cube and folded n-cube(n≥6),respectively,which both are distance-transitive graphs satisfying above assumption.At last,we will discuss the code upper bounds in the folded n-cube via its Terwilliger algebra and semidefinite programming.The main results are as follows:1.We obtain the Terwilliger algebra structures of one class of distance-transitive graph including a basis of this algebra and its dimension under Assumption 2.6.2.We first prove that,both bipartite half of n-cube and folded n-cube(n≥6)satisfy Assumption 2.6.We then,respectively,study the Terwilliger algebra structures of these two graphs including a basis of this algebra.the dimension of this algebra,the isomorphism classes of its irreducible modules and multiplicities the of its irreducible modules.3.For the folded n-cube,often denoted by □n,by using the theory of Terwillger algebra irreducible module we describe the block-diagonalization the Terwilliger algebra of □n.4.Based on block-diagonalizing the Terwilliger algebra of □n and on semidefinite programming we give an upper bound on A(□n,d)called semidefinite programming upper bound on A(□n,d)where A(□n,d)denotes the maximum size of a code with minimum distance at least d.Moreover,we prove that semidefinite programming upper bound on A(□n,d)≤Delsarte upper bound on A(□n,d).Furthermore,we offer several concrete upper bounds on A(□n,d)for 8≤n≤13.By comparison,we find that semidefinite programming upper bound on A(□n,d is at least as strong as Delsarte upper bound on that.