The Hadamard conjecture says that square matrices with ±1 entries and pairwise orthogonal rows exist in all orders 4n. The stronger Williamson conjecture says that Williamson matrices can be used to construct Hadamard matrices in all orders 4n.
The matrices we submitted to the Magma team can be found in the Hadamard matrix collection subpage. The scripts and instances we used to verify the nonexistence of Williamson matrices of order 35 can be found in the Williamson conjecture counterexample subpage. It was previously known that 35 is the smallest odd counterexample to the Williamson conjecture; we showed that it is in fact the smallest counterexample by constructing examples in all orders below 35, including those of even orders.
Also, it was previously unknown if eight-Williamson matrices exist in order 35 and we showed for the first time that eight-Williamson matrices do exist in order 35.
We have also constructed a new skew Hadamard matrix of order 4·57, found a new set of Williamson matrices in order 63, and new best matrices in order 57 (that can be used to construct skew Hadamard matrices of order 4·57).