Existence of a Hadamard matrix of order 668
Determine whether a real Hadamard matrix H ∈ M_{668}(±1) with pairwise orthogonal rows exists.
References
At higher $N$ things become more technical, and in order to discuss this, we will need:
\begin{theorem} The determinant of the $4\times4$ matrices is given by \begin{eqnarray*} &&\begin{vmatrix}a_1&a_2&a_3&a_4\ b_1&b_2&b_3&b_4\ c_1&c_2&c_3&c_4\ d_1&d_2&d_3&d_4\end{vmatrix}\ &=&a_1b_2c_3d_4-\ldots \end{eqnarray*} ... At higher $N$ things become more technical, and the continuation of it, we have:
\begin{conjecture}[Hadamard] There is an Hadamard matrix of order $N$, $$H\in M_N(\pm1)$$ for any $N\in4\mathbb N$. \end{conjecture} ... At higher $N$ things become more technical, and more complicated constructions, along the lines of those of Paley and Williamson, are needed. Quite curiously, as of now, early 21th century, the human knowledge stops at the number of the beast, namely: $$\mathfrak N=666$$ That is, explicit examples of Hadamard matrices have been constructed for all multiples of four $N\leq 664$, but no such matrix is known so far at $N=668$.