The Hadamard Maximal Determinant Problem (original) (raw)
Papers
- R. P. Brent, J. H. Osborn, W. P. Orrick and P. Zimmermann,
Maximal determinants and saturated D-optimal designs of orders 19 and 37,
submitted (currently under revision). Preliminary version available from arXiv 1112.4160v1. - R. P. Brent,
Finding D-optimal designs by randomised decomposition and switching,
Australasian Journal of Combinatorics 55 (2013), 15-30.
Also arXiv 1112.4671v5. Erratum. - R. P. Brent and J. H. Osborn,
General lower bounds on maximal determinants of binary matrices,
The Electronic Journal of Combinatorics 20(2), 2013, #P15, 12 pp.
Also arXiv:1208.1805v6. - R. P. Brent and J. H. Osborn,
Bounds on minors of binary matrices,
Bulletin of the Australian Math. Soc. 88 (2013), 280-285.
Also (longer version) arXiv:1208.3330v3. - R. P. Brent and J. H. Osborn,
On minors of maximal determinant matrices,
Journal of Integer Sequences 16 (2013), article 13.4.2, 30 pp.
Also arXiv:1208.3819v3. For relevant data files, see below. - R. P. Brent, J. H. Osborn and W. D. Smith,
Lower bounds on maximal determinants of +-1 matrices via the probabilistic method.
Available from arXiv:1211.3248v3, 5 May 2013.
[Largely superseded by the following.] - R. P. Brent, J. H. Osborn and W. D. Smith,
Lower bounds on maximal determinants of binary matrices via the probabilistic method.
Available from arXiv:1402.6817v2, 26 Jan. 2015. - R. P. Brent, J. H. Osborn and W. D. Smith,
Probabilistic lower bounds on maximal determinants of binary matrices,
Australasian Journal of Combinatorics 66 (2016), 350-364.
Also (slightly revised version) arXiv:1501.06235v7, 25 Oct. 2016.
For a longer version see arXiv:1402.6817v6.
Talks
Talks by Richard Brent
Workshop on Probabilistic and Extremal Combinatorics, Monash University, 13-17 June 2016
- Probabilistic lower bounds on maximal determinants of binary matrices
(joint work with Judy-anne Osborn and Warren Smith).
Overhead transparencies (beamer format): [pdf]
Gene Golub Day Memorial Workshop, Hong Kong Baptist University, 7 February 2015
- Lower bounds for the Hadamard maximal determinant problem
(joint work with Judy-anne Osborn and Warren Smith).
Overhead transparencies (beamer format): [pdf]
57th Annual Meeting of the Australian Mathematical Society, Sydney, 1 October 2013
- Improved lower bounds on the Hadamard maxdet problem, Part I
(joint work with and presented by Judy-anne Osborn).
Overhead transparencies (beamer format): [pdf]
Preprint: arXiv:1309.2795v2 - Improved lower bounds on the Hadamard maxdet problem, Part II
(joint work with Judy-anne Osborn and Warren Smith).
Overhead transparencies (beamer format): [pdf]
Preprint (March 2014): arXiv:1402.6817v2
36th ACCMCC Conference, University of NSW, 10-14 December 2012
- Lower bounds on maximal determinants via the probabilistic method
(joint work with Warren Smith and Judy-anne Osborn)
We use the probabilistic method to give improved lower bounds on the maximal determinant of n times n {+-1} matrices.
Overhead transparencies (beamer format): [pdf]
56th Annual Meeting of the Australian Mathematical Society, Ballarat, 26 September 2012
- Lower bounds on maximal determinants
(joint work with Judy-anne Osborn)
We give lower bounds on the maximal determinant of n times n {+-1} matrices, both with and without the assumption of the Hadamard conjecture.
Overhead transparencies (beamer format): [pdf]
Hadamard Workshop in Honour of Kathy Horadam, RMIT, Melbourne, 28 November 2011
- Finding many solutions of the Hadamard maximal determinant problem given the maximal Gram matrices
(presented at the International Workshop on Hadamard Matrices and their Applications in Honour of Kathy Horadam's sixtieth birthday, RMIT University, Melbourne, Australia, 28-30 November, 2011).
We describe how randomised decomposition and switching can be used to find many solutions of the Hadamard maximal determinant problem for certain orders.
Revised overhead transparencies (beamer format): [pdf]
Hadamard Maximal Determinant Workshop, ANU, 14 May 2010
- Decomposing Gram matrices,
(invited talk presented at the Hadamard Maximal Determinant Workshop, Australian National University, Canberra, 13-17 May 2010).
We discuss algorithms to decompose Gram matrices in order to find {+1,-1} matrices of maximal determinant.
Overhead transparencies (beamer format): [pdf] - See also my talks page.
Talks by Will Orrick and Paul Zimmermann
Data for Minors of Maxdet Matrices
The following links are relevant to the paper On minors of maximal determinant matrices.
In the pdf files, frequencies (multiplicities) of minors are indicated by superscripts. For example, 2336 means that the minor with absolute value 2 occurs 336 times.
- Minors and frequencies for n = 0 mod 4, n ≤ 16: pdf; n = 20: txt.
- Minors and frequencies for n = 1 mod 4, n ≤ 17: pdf.
- Minors only for n = 21: txt.
- Minors and frequencies for n = 2 mod 4, n ≤ 18: pdf.
- Minors and frequencies for n = 3 mod 4,n ≤ 15: pdf; n = 19: pdf.
Data for Various Orders
Gram matrices are encoded in a compressed format, which is described here.
A C program to convert to compressed format is here, and a C program to convert from compressed format is here.
The following links are relevant to the orders 19, 26, 27, 29, 33 and 37 that are considered in the above papers/talks.
Order 19
Order 26
Order 27
Order 29
Order 33
Order 37
Other Online Resources
A great resource is Will Orrick's page.