Hadamard matrix existence for all multiples of four

Determine whether Hadamard matrices exist for every order m that is a multiple of four (m = 4k), thereby resolving the Hadamard matrix conjecture.

Background

The paper recalls that Hadamard matrices have orders 1, 2, or multiples of 4, but their existence for every multiple of 4 remains unresolved. This classical problem influences constructions and bounds related to maximal simplices in cubes and interpolation projector norms.

Several results in the paper (e.g., exact bounds tied to Hadamard numbers) depend on whether an Hadamard matrix of a given order exists.

References

But it is still unknown whether an Hadamard matrix exists for every order of the form m=4k. This is one of the longest lasting open problems in Mathe-matics called the Hadamard matrix conjecture.

— Optimal Lagrange Interpolation Projectors and Legendre Polynomials  (2405.01254 - Nevskii, 2024) in Section 2 (Notation and preliminaries), after Definition 2.13 (Hadamard matrices)

A famous open problem with Hadamard matrices is the existence of such matrices for all multiples of 4, with currently 768 being the currently smallest such order that no known Hadamard matrices have been found yet ( resolved the previously lower bound of 764 in 2008).

— Complete pivoting growth of butterfly matrices and butterfly Hadamard matrices  (2410.06477 - Peca-Medlin, 2024) in Section 1 (Introduction)

Their existence for all such $N$ is a famous open problem .

— New constructions of optimal arrangements of $2d$ lines in $\mathbb{C}^d$  (2608.16116 - Glazyrin, 17 Aug 2026) in Section 2, Hadamard and conference matrices