The Existence Of 1 (1/2)-designs And The Constructions Of Orthogonal Arrays

In this paper,we study the existence problem of two kinds of combinatorial config-urations:1 (1/2)-designs and orthogonal arrays with strength 3.The first research content of this paper is 1 (1/2)-design.t1/2-design is the generalization of classical t-design,which is closely related to directed strongly regular graphs and other combinatorial configurations.In 1980,Neumaier first proposed the concept of t1/2-design,and gave the classification of t,1/2-designs with t>2.Therefore,in recent years,the research on t1/2-designs mainly focuses on the case of 1 (1/2)-designs.For fixed blocksize k,the necessary and sufficient conditions for the existence of 1 (1/2)-designs with blocksize 3 have been given in the literature.In this paper,we continue to study the designs with blocksize 4.By establishing the incidence matrix for the blocks containing a given point,we get the relationship between different parameters,and prove the nonexistence of a class of 1 (1/2)-designs of type[3,2,2].The second research content of this paper is orthogonal array.Orthogonal array is a combinatorial structure introduced by Rao in the study of experimental designs in 1947.It has an important application in statistics,computer science,coding theory,cryptography and other fields.The orthogonal array with strength 3 has always been a very difficult but fascinating topic in combinatorial design theory.For the 6-row orthogonal arrays of order 3p(prime p?5)with strength 3,the literature only gives the construction results for p=5 and 7.In this paper,by designing an efficient computer search algorithm,a 4-row difference matrix of order 33 with an adder is obtained,and the existence of a 6-row orthogonal array of order 33 with strength 3 is determined.By using the known recursive construction,some new infinite classes of orthogonal arrays with strength 3 are given.