Combination Scheme Of Hermitian Complex Hadamard Matrix And 4-type Oblique Symmetry

The association scheme was originally a combined structure accompanied by a partially balanced incomplete block design,It was introduced in 1952 by Bose and Shimamoto.With the development of research in recent years,associative schemes have played an important role in many branches of mathematics.With the development of graph theory,coding,design and finite geometry,associative scheme has become an important branch of algebraic combinatorial theory.The results of the integration programme are mainly compiled in books by Bannai and Ito,published in 1984.Hadamard matrix has a wide range of applications in experimental design,coding,network,logic circuit and other aspects,and can be constructed in a variety of ways.The application of Hadamard matrix can be divided into theoretical application and practical application.Its theoretical applications include polyhedron theory,coding theory and game theory,etc.Its practical applications include network theory,logic circuit theory,automata theory,Fourier spectrum analysis,Walsh function and so on.In associative scheme theory,associative schemes with a small number of classes are closely related to strongly regular graphs.In the 70 years since the joint schemes were proposed,symmetric joint schemes with small class numbers have been extensively studied.1-class joint schemes are equivalent to complete graphs.2-class symmetric joint schemes are equivalent to a pair of complementary strongly regular graphs.The concept of strongly regular graphs has been introduced for nearly 60 years.For the content of strongly regular graphs,see Hubaut’s 1975 review paper.In this paper,we use strongly regular graphs and 2-class symmetric associations to split the scheme into 4-class skew-symmetric associations,we consider Hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of 4-class skewsymmetric association schemes.We give a characterization of the character table of a 4-class skew-symmetric association scheme whose Bose-Mesner algebra contains a Hermitian complex Hadamard matrix.