Nonexistence conjecture for negacyclic Hadamard matrices

A negacyclic Hadamard matrix is a Hadamard matrix whose rows are generated from one another by negacyclic shifts. Negacyclic Hadamard nonexistence conjecture. There are no negacyclic Hadamard matrices of order $>2$. This is the negacyclic analogue of the conjecture attributed to Ryser that there are no cyclic Hadamard matrices of order $>4$. The authors verified the assertion computationally for orders at most $40$, but it is not established in general.