While the order of a Hadamard matrix must be 1, 2, or a multiple of 4, regular Hadamard matrices carry the further restriction that the order must be a square number.
If n = 4u 2 is the order of a regular Hadamard matrix, then the excess is ±8u 3 and the row and column sums all equal ±2u.
If H is interpreted as the incidence matrix of a block design, with 1 representing incidence and −1 representing non-incidence, then H corresponds to a symmetric 2-(v,k,λ) design with parameters (4u 2, 2u 2 ± u, u 2 ± u).
A number of methods for constructing regular Hadamard matrices are known, and some exhaustive computer searches have been done for regular Hadamard matrices with specified symmetry groups, but it is not known whether every even perfect square is the order of a regular Hadamard matrix.
Bush-type Hadamard matrices are regular Hadamard matrices of a special form, and are connected with finite projective planes.