Menon Hadamard Difference Sets and Regular Hadamard Matrices

This chapter deals with Menon Hadamard difference sets and regular Hadamard matrices. The existence of a (4u
²
,2u
²
±u,u
²
±u) Menon Hadamard difference set is equivalent to the existence of a regular Hadamard matrix of order 4u
²
. Turyn proved the existence of a family of Hadamard matrices. Xia constructed a family of Menon Hadamard difference sets when u is an odd prime power. Later Xiang–Chen gave a character theoretic proof based on cyclotomy. The chapter presents a theorem which determines cyclotomic periods explicitly for special cases and is the basis for the proof of Xia's construction. Wilson and Xiang extended Xia's construction of Menon Hadamard difference sets based on projective geometry. The chapter also presents the construction based on Galois rings.