Hadamard Matrix of Order 668

There exists a Hadamard matrix of order $668$: a matrix $$ H\in\{-1,1\}^{668\times668} $$ such that $$ HH^{\mathsf T}=668I_{668}. $$ Equivalently, the $668$ rows of $H$ are pairwise orthogonal.

Explicit construction of a Hadamard matrix of order 668, the smallest previously unresolved order, verified exactly by this site from the announcement plus its decoder reply. The same post encodes matrices for all twelve previously-open admissible orders below 2000 (668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964), and this site verified every one of them. The entry records the order-668 existence question, which this fully resolves; the general Hadamard conjecture - existence for ALL admissible orders - remains open, with the smallest unknown order now 2004 or beyond.