Since early 2002, Bruce Solomon and I have maintained an online database of record-determinant {1,−1}-matrices.  Around that time Bruce was setting lots of new records by means of computer search techniques that grew out of his collaboration with Roland Dowdeswell, Michael Neubauer, and Kagan Tumer, using some key ideas that had been developed by Warren Smith in his 1988 Princeton University PhD dissertation.  Trevor Welsh was also at that time establishing many new records for larger sizes congruent to 3 (mod 4), and, a short while later, I succeeded in proving maximality of Smith’s conjectured 15×15 maximal-determinant matrix.  The website was a way to keep the community apprised of the rapid progress that was then being made.   (There had been an earlier site, developed by Roland Dowdeswell, that was connected to a permanently running search program.  That site appears to have been taken offline.)  Eventually progress slowed, and the day-to-day maintenance of the site fell to me.

In 2005 there was another spate of new records, set by the participants in the programming contest organized by Lars Backstrom.  I regret that, while I continued to perform sporadic updating of the site, the records for certain matrix sizes have remained out-of-date since that time.

The reason for this post is to announce that, finally, the page has been brought up-to-date.  The sizes that are affected by this revision are n=45 in the 1 (mod 4) column, n=70, 78, and 94 in the 2 (mod 4) column, and n=35, 39, 43, and 63 in the 3 (mod 4) column.  My apologies to anyone who may have worked to improve any of the records that were out-of-date.  I will strive to keep the page more current in the future.  Even though the main page is up-to-date, many of the subsidiary pages for particular n are still out of date.  In particular, there are a number of classification results that have not yet made it to those pages.

I have plans to write a series of follow-up posts explaining the developments that led to these improvements.   Here’s a brief summary:

  • the methods using “doubly three-normalized” Hadamard matrices described in Orrick, W. P. and Solomon, B., Large-determinant sign matrices of order 4k+1, Discrete Math. 307 (2007) 226–236.  (arXiv preprint) were used to find the new record for n=45 by Brent, Orrick, Osborne, and Zimmermann.  The result was reported, but not described in detail, in http://arxiv.org/abs/1112.4160.
  • the results for n=70, 78, and 94 have been available at Tomas Rokicki’s website since 2005.  All are constructed from pairs of bordered, circulant matrices and were found by Rokicki or by Orrick and Rokicki.  These results have not been published, but a paper exists in draft form.  Additional information about the structure of the record for n=70, which is similar to the structure of the records for n=22, 34, and 106, can be found in the REU report of Adam Vollrath.
  • the results for n=35, 39, 43, and 63 were found and reported to the site by Hiroki Tamura in January 2011.  They are related to the work in Tamura, Hiroki, D-optimal designs and group divisible designs, Journal of Combinatorial Designs 14 (2006) 451–462.  Similar results for n=27, 31, and 59 were reported to the site by Tamura back in August 2005 (and added to the site at that time).  Apparently none of these results have yet been published, but a paper may be in the works.
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