Status of Hadamard matrix conjecture I  would  like  to  know  if  any  progress  has  been  made  on Hadamard conjecture : 

Hadamard matrix of order $4k$ exists for every positive integer $k$.

 A: With all due respect to Colin McLarty, here is (in my not so humble opinion) a better answer.
The conjecture (For every positive integer $k,$ there is a square matrix $H$ of order $4k$ such that $H$ is binary with entries being $1$ or $-1$, with $HH^t = 4k~I$) is not resolved, but there is much work going on in the area.


*

*The conjecture is generalized to the Hadamard maximum determinant problem (with binary matrices of all orders, not just $4k$), which in turn is generalized to the determinant spectrum problem (range of determinant function over binary matrices of a given order, a personal favorite of mine).  Will Orrick helps maintain a website http://indiana.edu/~maxdet presently for these problems.  My motivation for working on the determinant spectrum problem came from attempting an oblique approach to HMC.

*when extended to complex entries of modulus 1, such a matrix exists for every order.  Such matrices are studied for their own interest, but many who peruse this part of the literature and say they are doing this without a thought to the (real number version of the) Hadamard matrix conjecture, well, let me stop short of name-calling: I don't believe that statement for a moment.

*other extensions concern combinatorial designs, partial Hadamard matrices, equivalence under a variety of relations, and ways of generating representatives.
If one takes off the blinders and asks what progress on and around the Hadamard Matrix Conjecture is being made, the Wikipedia and Mathworld articles mentioned in other posts are a good start.  In addition to the maxdet website and similar websites, the ArXiv has at least 10 articles from the last few years on recent work.
Gerhard "Progress Isn't Just A Number" Paseman, 2017.11.02.
A: According to the current version on the MathWorld website by Wolfram,
http://mathworld.wolfram.com/HadamardMatrix.html :
"… the smallest unknown order [$of\ a\ possible\ Hadamard\ matrix$] is 668."
I suppose this is what you wanted to know. In addition, the website provides important references on this subject.
A: According to Wikipedia (last edited on 31 March 2017, at 03:48.) the Hadamard conjecture is open still.
