On the Existence of Hadamard Matrices

Given any natural number q > 3 we show there exists an integer t ≤ [2 log2 (q – 3)] such that an Hadamard matrix exists for every order 2q where s > t. The Hadamard conjecture is that s = 2. This means that for each q there is a finite number of orders 2q for which an Hadamard matrix is not known. This is the first time such a statement could be made for arbitrary q. In particular it is already known that an Hadamard matrix exists for each 2q where if q = 2 – 1 then s ≥ m, if q = 2 + 3 (a prime power) then s ≥ m, if q = 2 + 1 (a prime power) then s ≥ m + 1. It is also shown that all orthogonal designs of types (a, b, m – a – b) and (a, b), 0 ≤ a + b ≤ m, exist in orders m = 2 and 2 ∙ 3, t ≥ 1 a positive integer.