Submit progress

on Smallest unsolved instance of the Hadamard conjecture ($C_{23a}$)

The statement

A Hadamard matrix of order $n$ is an $n\times n$ matrix $H$ with entries in $\{-1,1\}$ such that

HH^{\top} = n I_n.

Equivalently, the rows (and columns) are pairwise orthogonal.

It is known that Hadamard matrices can exist only for $n=1,2$ , or $n\equiv 0 \pmod{4}$ .

We define $C_{23a}$ to be the smallest integer $n\equiv 0 \pmod{4}$ such that there is no Hadamard matrix of order $n$ . If no such $n$ exists, we set $C_{23a}=\infty$ .

The Hadamard conjecture asserts that $C_{23a}=\infty$ , i.e. that Hadamard matrices exist for every order $n\equiv 0 \pmod{4}$ .

(Equivalently, by Hadamard’s determinant inequality, for $A\in\{-1,1\}^{n\times n}$ one has $|\det(A)|\le n^{n/2}$ , with equality iff $A$ is Hadamard; the conjecture predicts equality is attainable for all $n\equiv 0\pmod4$ .)

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.