Partial Geometric Designs With Block Size 5

In 1963,Bose proposed the definition of partial geometry.In 1976,Bose et al.promot-ed partial geometry and gave the concept of partial geometric design which was equivalent to the definition of 11/2-design proposed by Neumaier in 1980.At the same time,Neumaier pointed out the relationship between 2-designs,transversal designs,symmetric complete bipartite graphs,partial geometries and partial geometric designs.Later,Brouwer et al.constructed directed strongly regular graphs by using flag and antiflag of partial geomet-ric designs.Therefore,more and more mathematics are devoted to the research on partial geometric designs.In 2015,Feng,Zeng et al.used the geometry of classical groups over finite fields to construct partial geometric design in symplectic geometry,unitary geometry and orthogo-nal geometry.In 2016,Chen classified some partial geometric designs with block size three and gave some relationships among their parameters.In 2018,Wang studied the partial geometric designs with block size four and gave the existence of some partial geometric designs with specific parameters.In 2020,Sun discussed the existence of partial geometric designs with block size four and type[3,2,2].In 2021,Qu,Lei and Shan discussed the existence spectrum of partial geometric designs with block size four and type[λ,1,1].This paper mainly studies the existence of partial geometric designs with block size five and type[1,λ,2λ-1].Combined with the necessary conditions of the existence of partial geometric designs,we analyse the block structures of this kind of partial geometric designs.Then,we give some important properties related to the parameters of this kind of partial geometric designs,and finally get the parameters of this type of partial geometric design,which comes down to a class of parameters and six specific parameters.