Hadamard Conjecture (Existence for all orders divisible by 4)

Establish the existence, for every positive integer N divisible by 4, of a real Hadamard matrix H ∈ M_N(±1) whose rows are pairwise orthogonal.

Background

An Hadamard matrix is a square binary matrix H ∈ M_N(±1) whose rows are pairwise orthogonal. Classical constructions (Walsh matrices via tensor powers and Paley-type constructions) produce examples for many sizes, and it is known that such matrices can only exist when N is 2 or a multiple of 4. The Hadamard Conjecture asserts existence for every N ∈ 4ℕ and has been a central open problem for more than a century.