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The following feels like a community wiki question, so I do it here:

Recently we have heard of a new proof of the Circulant Hadamard conjecture of Ryser (a long standing difficult conjecture):

There is no Circulant Hadamard matrix with more than $4$ columns

see

Barry Hurley, Paul Hurley, Ted Hurley, The Hadamard circulant conjecture, Bulletin of the London Mathematical Society, Volume 44, Issue 1, February 2012, Page 206, https://doi.org/10.1112/blms/bdq112

More recently a gap was found in the proof:

see

https://www.sfu.ca/~jed/Papers/Craigen%20Jedwab.%20Circulant%20Hadamard.%20Preprint.pdf

Question: What is the real status of this Conjecture ?

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  • $\begingroup$ This is not my field, but I don't know of any progress since the case of 448 was done. $\endgroup$
    – Kimball
    Commented Feb 15, 2011 at 16:17
  • $\begingroup$ @Kimball: Your comment concerns the Hadamard Conjecture: For each posotive integer $n$ multiple of $4$ there exist an Hadamard matrix with $n$ columns. For both conjectures see the web wiki page that gives details on both, but not enough... $\endgroup$
    – Luis H Gallardo
    Commented Feb 15, 2011 at 18:35
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    $\begingroup$ Given both articles are from 2011, I'd say you are as on top of the status as anyone could be. Robert Craigen is approachable and (with the possible exception of Will Orrick) would be the first person I would ask about the conjecture. Gerhard "Ask Me About System Design" Paseman, 2011.02.15 $\endgroup$
    – Gerhard Paseman
    Commented Feb 15, 2011 at 19:53
  • $\begingroup$ Kimball, if you mean 428, indeed, no new Hadamard matrices with new orders have been announced to me in the last 7 years. If you mean 448 (= 28x16), you might check some (almost any) combinatorial design literature from the 20th century. Gerhard "Slow, But Not That Slow" Paseman, 2011.02.15 $\endgroup$
    – Gerhard Paseman
    Commented Feb 16, 2011 at 4:14
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    $\begingroup$ It has been circulating around.. $\endgroup$
    – Qfwfq
    Commented Feb 16, 2011 at 23:59

7 Answers 7

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Unfortunatly, there has been confusion about the status of this problem for decades.

All correct results on the circulant Hadamard matrix conjecture are contained in the following papers:

[1] Turyn, Richard J.: Character sums and difference sets. Pacific J. Math. 15 1965 319–346.

[2] Leung, Ka Hin; Schmidt, Bernhard: The field descent method. Des. Codes Cryptogr. 36 (2005), no. 2, 171–188.

[3] Leung, Ka Hin; Schmidt, Bernhard: New restrictions on possible orders of circulant Hadamard matrices. Des. Codes Cryptogr. 64 (2012), no. 1-2, 143–151.

Computational results and the latest status on the smallest open cases can be found in

[4] Peter Borwein, Michael J. Mossinghoff: Wieferich pairs and Barker sequences, II. arXiv:1306.0045v2

Bernhard Schmidt

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The paper by Craigen and Jedwab points out a very definite flaw in the main theorem by Hurley, Hurley & Hurley, providing a counterexample to that theorem. So the conjecture is still open.

The paper by Leung and Schmidt (here) gives the latest information about possible counterexamples to the Circulant Hadamard conjecture.

A website of Bernhard Schmidt lists smallest open cases based (I believe) on the work of the paper cited above. The smallest open cases are, of course, quite large.

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Craigen and Jedwab have indicated that the September 2011 revision of Hurley, Hurley & Hurley also has a flaw:

https://www.sfu.ca/~jed/Papers/Craigen%20Jedwab.%20Circulant%20Hadamard%202.%20Preprint.pdf

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About two weeks ago, at the International Workshop on Hadamard Matrices and Their Applications (RMIT), http://user.gs.rmit.edu.au/asha/iwhma/ Jennifer Seberry presented a proof, based on counting arguments http://user.gs.rmit.edu.au/asha/iwhma/html/Wednesday/JenniferSeberry.pdf (Wayback Machine), but that proof was quickly found to be incomplete. As far as I know, she is currently working on improving the proof. A special volume of the Australasian Journal of Combinatorics http://user.gs.rmit.edu.au/asha/iwhma/html/ajc.html is forthcoming, and with any luck, things will become much clearer at that point.

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  • $\begingroup$ this complements Penguian's post: uow.edu.au/~jennie/11pub.html gives: JOURNAL ARTICLES SUBMITTED Lei Wang, Tianbing Xia and Jennifer Seberry, UMAC for identifying IP spoofing attacks, (submitted) Tianbing Xia, Jennifer Seberry and Mingyuan Xia, Some new constructions for orthogonal designs, (submitted) Jennifer Seberry, Non-existence of circulant Hadamard matrices for orders $>$ 4 (submitted BLMS 17/11/11) Jennifer Seberry, New families of amicable Hadamard matrices, clicking on "Non-existence of circulant Hadamard matrices for orders $>$ 4 " you get: $\endgroup$
    – Luis H Gallardo
    Commented Dec 19, 2011 at 22:39
  • $\begingroup$ continued here: You went from this referring page: uow.edu.au/~jennie/11pub.html To this missing page: uow.edu.au/~jennie/WEB/WEB11/no-circ.17Nov.pdf $\endgroup$
    – Luis H Gallardo
    Commented Dec 19, 2011 at 22:40
  • $\begingroup$ Now, february 2012, above address changed contents to: (in prep) i.e., now seems that the paper, that has order number 10, become ``in preparation''... $\endgroup$
    – Luis H Gallardo
    Commented Feb 14, 2012 at 16:59
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On sep 4, 2011, authors Hurley, Hurley & Hurley of the paper on the Circulant Hadamard conjecture, has posted the same paper on the arxiv. Did they corrected the error reported by Craigen and Jedwab on feb 10, 2011?

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  • $\begingroup$ If you give a link, then this might help people to answer $\endgroup$
    – Yemon Choi
    Commented Sep 29, 2011 at 23:36
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    $\begingroup$ arxiv.org/pdf/1109.0748v1 $\endgroup$
    – Chris Godsil
    Commented Sep 30, 2011 at 0:07
  • $\begingroup$ @Chris: thanks, I now vaguely recall seeing this in the daily listings at the time. I note that the new/revised version makes no explicit reference that I can see to the earlier effort, nor to the Craigen and Jedwab paper $\endgroup$
    – Yemon Choi
    Commented Sep 30, 2011 at 1:04
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    $\begingroup$ @Yemon: actually they state that it's a "post publication revision of on-line Bull. London Math. Soc. version", which is very delicately phrased. The lack of a reference to Craigen and Jedwab is a little strange. $\endgroup$
    – Chris Godsil
    Commented Sep 30, 2011 at 12:20
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I recently found out about this conjecture and found this paper from 2019 on arXiv,

https://arxiv.org/pdf/1907.12683.pdf

As far as I can tell, this paper is still unpublished and on the Wikipedia-page, it says that Circulant Matrix Conjecture is still an open problem. Is the proof in the aforementioned paper known to have gaps?

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    $\begingroup$ for what its worth, there's already a grammatical typo in the first page, and more appear in the next few pages. while that doesn't refute the paper at all, is is a red flag to me. $\endgroup$
    – Zach Hunter
    Commented Jan 10, 2021 at 16:47
  • $\begingroup$ Some people like to look at the abstract, before deciding whether to load the pdf: arxiv.org/abs/1907.12683 $\endgroup$
    – Gerry Myerson
    Commented Jan 11, 2021 at 0:39
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check:

Barry Hurley, Paul Hurley, Ted Hurley, The Hadamard circulant conjecture, Bulletin of the London Mathematical Society, Volume 44, Issue 1, February 2012, Page 206, https://doi.org/10.1112/blms/bdq112

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