Symmetric and skew Hadamard matrices in every order divisible by four

Construct both symmetric and skew Hadamard matrices for every order divisible by 4.

Background

The paper distinguishes symmetric Hadamard matrices from skew Hadamard matrices, the latter having the form C + I with C skew-symmetric. It records the widely held belief that both types should exist in every order divisible by 4, but does not establish this universal existence claim.

References

Both symmetric and skew Hadamard matrices of order $N$ are widely believed to exist for all $N$ divisible by 4 (see for the current status of all these conjectures).

— 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