I would like to know if any progress has been made on Hadamard conjecture :
Hadamard matrix of order $4k$ exists for every positive integer $k$.
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4$\begingroup$ The wikipedia article seems up to date. $\endgroup$– Chris GodsilCommented Jan 8, 2012 at 18:48
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7$\begingroup$ Warwick de Launey's paper "On the Asymptotic Existence of Hadamard Matrices", arxiv.org/abs/1003.4001, may be of interest. $\endgroup$– Will OrrickCommented Jan 8, 2012 at 19:45
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3 Answers
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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.
answered Nov 2, 2017 at 20:04
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4$\begingroup$ You might also note that for most large k we do not know how to construct a Hadamard matrix of order 4k; the fact that the smallest open case is 167 is misleading. The density of k for which we know a Hadamard matrix decays as c/log(k); for example, one construction assumes 4k-1 is prime. $\endgroup$ Commented Nov 2, 2017 at 20:40
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$\begingroup$ @Noam, except I disagree with this evaluation of most also. My metric for odd k is t, the power of 2 for which we know a matrix of order k2^t exists. The reference posted by Orrick above is recent. When I find something more recent (or understand the effectiveness of Keevash's recent work), I will post more. Gerhard "Progress Can Be A Function" Paseman, 2017.11.02. $\endgroup$ Commented Nov 2, 2017 at 22:03
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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.
answered Nov 2, 2017 at 19:05
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According to Wikipedia (last edited on 31 March 2017, at 03:48.) the Hadamard conjecture is open still.
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5$\begingroup$ It is useful to have this information here now but, in the long run, I don't think having such answers makes much sense. I mean why now? Do you mean to update it, say, every month, or every year, or what? $\endgroup$ Commented Nov 2, 2017 at 14:21
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2$\begingroup$ @მამუკაჯიბლაძე This is a problem with the question. It does not admit any better answer. $\endgroup$ Commented Nov 2, 2017 at 19:10