Lower bounds on maximal determinants of +-1 matrices via the probabilistic method

Brent, Richard P.; Osborn, Judy-anne; Smith, Warren D.

doi:10.48550/arxiv.1211.3248

Cited by 4 publications

(6 citation statements)

References 36 publications

(60 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The same approach gives weak results for d > 3 because of the large number (d! − 1) of off-diagonal terms (see [6,Theorem 1]). Theorem 3.6.…”

Section: A Probabilistic Lower Boundmentioning

confidence: 99%

General Lower Bounds on Maximal Determinants of Binary Matrices

Brent¹,

Osborn²

2013

Electron. J. Combin.

Self Cite

View full text Add to dashboard Cite

Let D(n) be the maximal determinant for n × n {±1}-matrices, and R(n) = D(n)/n n/2 be the ratio of D(n) to the Hadamard upper bound. Using the probabilistic method, we prove new lower bounds on D(n) and R(n) in terms of d = n − h, where h is the order of a Hadamard matrix and h is maximal subject to h ≤ n. For example,By a recent result of Livinskyi, d 2 /h 1/2 → 0 as n → ∞, so the second bound is close to (πe/2) −d/2 for large n. Previous lower bounds tended to zero as n → ∞ with d fixed, except in the cases d ∈ {0, 1}. For d ≥ 2, our bounds are better for all sufficiently large n. If the Hadamard conjecture is true, then d ≤ 3, so the first bound above shows that R(n) is bounded below by a positive constant (πe/2) −3/2 > 0.1133.

show abstract

“…The same approach gives weak results for d > 3 because of the large number (d! − 1) of off-diagonal terms (see [6,Theorem 1]). Theorem 3.6.…”

Section: A Probabilistic Lower Boundmentioning

confidence: 99%

General Lower Bounds on Maximal Determinants of Binary Matrices

Brent¹,

Osborn²

2013

Electron. J. Combin.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The right-handside of eqn. (10), which by [9, Corollary 3.1] is a lower bound on the number of vanishing minors in this non-Hadamard case, gives 1362 (rounded up).…”

Section: Further Results and Observations On Minorsmentioning

confidence: 99%

“…Remark 3. It should be possible to sharpen Theorems 2-3 by using the (asymptotically sharper) bounds on D(m) given in [10] instead of the bound of Proposition 1 (this is work in progress).…”

Section: Excluded Minors Of Hadamard Matricesmentioning

confidence: 99%

On minors of maximal determinant matrices

Brent,

Osborn

2012

Preprint

Self Cite

View full text Add to dashboard Cite

By an old result of Cohn (1965), a Hadamard matrix of order n has no proper Hadamard submatrices of order m > n/2. We generalise this result to maximal determinant submatrices of Hadamard matrices, and show that an interval of length ∼ n/2 is excluded from the allowable orders. We make a conjecture regarding a lower bound for sums of squares of minors of maximal determinant matrices, and give evidence to support it. We give tables of the values taken by the minors of all maximal determinant matrices of orders ≤ 21 and make some observations on the data. Finally, we describe the algorithms that were used to compute the tables.

show abstract

“…In Corollary 1 we relax the condition on a ii to a one-sided constraint a ii ≥ 1−δ. The results have applications to proofs of lower bounds for the Hadamard maximal determinant problem; this was our original motivation (see [4,5]). Regarding other reasons for considering bounds on determinants, we refer to Bornemann [3, footnote 4].…”

Section: Introductionmentioning

confidence: 99%

Bounds on determinants of perturbed diagonal matrices

Brent,

Osborn,

Smith

2014

Preprint

Self Cite

View full text Add to dashboard Cite

We give upper and lower bounds on the determinant of a perturbation of the identity matrix or, more generally, a perturbation of a nonsingular diagonal matrix. The matrices considered are, in general, diagonally dominant. The lower bounds are best possible, and in several cases they are stronger than well-known bounds due to Ostrowski and other authors. If A = I − E is a real n× n matrix and the elements of E are bounded in absolute value by ε ≤ 1/n, then a lower bound of Ostrowski (1938) is det(A) ≥ 1 − nε. We show that if, in addition, the diagonal elements of E are zero, then a best-possible lower bound is Corresponding upper bounds are respectivelyThe second upper bound generalises Hadamard's inequality, which is the case ε = 1. A necessary and sufficient condition for our upper bounds to be best possible for matrices of order n and all positive ε is the existence of a skew-Hadamard matrix of order n.

show abstract

Lower bounds on maximal determinants of +-1 matrices via the probabilistic method

Cited by 4 publications

References 36 publications

General Lower Bounds on Maximal Determinants of Binary Matrices

General Lower Bounds on Maximal Determinants of Binary Matrices

On minors of maximal determinant matrices

Bounds on determinants of perturbed diagonal matrices

Contact Info

Product

Resources

About