Lethbridge Number Theory and Combinatorics Seminar: Hadi Kharaghani

A Hadamard matrix H of order 4n2 is said to be skew-regular if it is of skew-type and the absolute values of the row sums are all 2n. It is conjectured that for each odd integer n there is a skew-regular matrix of order 4n2.

A Hadamard matrix H of order m is said to be balancedly splittable if there is an ℓ×m submatrix H1 of H such that inner products for any two distinct column vectors of H1 take at most two values. It is conjectured that only (Sylvester) Hadamard matrices of order 4n are balancedly splittable.

The existence and applications of these two very interesting classes of matrices to Hadamard diagonalizable strongly regular graphs, maximal equiangular lines set, doubly regular tournament, and unbiased Hadamard matrices will be discussed in detail.