The excess of Hadamard matrices and optimal designs

Farmakis, Nikolaos; Kounias, Stratis

doi:10.1016/0012-365x(87)90025-2

Cited by 21 publications

(20 citation statements)

References 9 publications

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“…; c n Þj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Since the bound is met only by regular Hadamard matrices, which thus occur only in square orders, and is known to be met in all orders n 2 , where n is the order of a Hadamard matrix, there remains the nontrivial question of maximum excess in nonsquare orders and square orders not divisible by 16. This question has attracted some attention over the last two decades (see, e.g., [10,11,13,16,17,18]). The actual value of the maximum excess is also known for a few infinite classes (see [13,15,17]).…”

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confidence: 98%

Weaving hadamard matrices with maximum excess and classes with small excess

Craigen

Kharaghani

2004

J of Combinatorial Designs

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Weaving is a matrix construction developed in 1990 for the purpose of obtaining new weighing matrices. Hadamard matrices obtained by weaving have the same orders as those obtained using the Kronecker product, but weaving affords greater control over the internal structure of matrices constructed, leading to many new Hadamard equivalence classes among these known orders. It is known that different classes of Hadamard matrices may have different maximum excess. We explain why those classes with smaller excess may be of interest, apply the method of weaving to explore this question, and obtain constructions for new Hadamard matrices with maximum excess in their respective classes. With this method, we are also able to construct Hadamard matrices of near-maximal excess with ease, in orders too large for other by-hand constructions to be of much value. We obtain new lower bounds for the maximum excess among Hadamard matrices in some orders by constructing candidates for the largest excess. For example, we construct a Hadamard matrix with excess 1408 in order 128, larger than all previously known values. We obtain classes of Hadamard matrices of order 96 with maximum excess 912 and 920, which demonstrates that the maximum excess for classes of that order may assume at least three different values. Since the excess of a woven Hadamard matrix is determined by the row sums of the matrices used to weave it, we also investigate the properties of row sums of Hadamard matrices and give lists of them in small orders.

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confidence: 98%

Weaving hadamard matrices with maximum excess and classes with small excess

Craigen

Kharaghani

2004

J of Combinatorial Designs

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“…In [16], Kounias and Farmakis derive an upper bound smaller than n 3/2 when n is not a perfect square, and tabulate their bound versus the largest known excess for all orders up to n = 100 [13]. For convenience, we reproduce their list of bounds in Table 1.…”

Section: The N 3/2 Bound On Maximal Excessmentioning

confidence: 97%

“…354.44 0.80 n+1 encodes the largest known determinant of order n + 1. Compare n+1 with the corresponding factor for our determinant (((n + n ) + n)/4), and that of [13] (( n + n)/4).…”

Section: Upper Boundsmentioning

confidence: 97%

“…Indeed, this constitutes the maximal excess technique of Farmakis and Kounias [13] we mention in our Introduction. By comparison, Theorem 3.1 above shows we can improve on that technique in any dimension n where there exists a 3-normalized n × n Hadamard matrix N with…”

Section: Upper Boundsmentioning

confidence: 97%

“…In these orders, the maximal excess construction of Farmakis and Kounias [13] gives a good general method for constructing determinants that reach a large fraction of B(n). For certain orders such as n = 13, 21, 25, 41, and 61, however, determinants beyond those attainable via maximal excess have been known for some time [22,8,[4][5][6].…”

Section: Overviewmentioning

confidence: 99%

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Large-determinant sign matrices of order 4k+1

Orrick

Solomon

2007

Discrete Mathematics

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The Hadamard maximal determinant problem asks for the largest n × n determinant with entries ±1. When n ≡ 1(mod 4), the maximal excess construction of Farmakis and Kounias [The excess of Hadamard matrices and optimal designs, Discrete Math. 67 (1987) 165-176] produces many large (though seldom maximal) determinants. For certain small n, still larger determinants have been known; e.g., see [W.P. Orrick, B. Solomon, R. Dowdeswell, W.D. Smith, New lower bounds for the maximal determinant problem, arXiv preprint math.CO/0304410]. Here, we define "3-normalized" n × n Hadamard matrices, and construct large (n + 1) × (n + 1) determinants from them. Our constructions give most of the previous "small n" records, and set new records when n = 37, 49, 65, 73, 77, 85, 89, 93, 97, and 101, most of which exceed the reach of the maximal excess technique. We suspect that our n = 37 determinant, 72 × 9 17 × 2 36 is best possible.

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Direct construction of globally D‐optimal designs for factors at two levels and main effects models

King

Jones

Morgan

et al. 2020

Quality & Reliability Eng

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In developing screening experiments for two‐level factors, practitioners typically are familiar with regular fractional factorial designs, which are orthogonal, globally D‐optimal (ie, 100% D‐efficient), and exist if N is a power of two. In addition, nonregular D‐optimal orthogonal designs can be generated for almost any N a multiple of four, the most notable being the family of Plackett and Burman1 designs. If resource constraints dictate that N is not a multiple of four, while an orthogonal design for two‐level factors does not exist, one can still consider a D‐optimal design. Exchange algorithms are available in commercial computer software for creating highly D‐efficient designs. However, as the number of factors increases, computer searches eventually fail to find the globally optimal design for any N or require impractical search times. In this article, we compile state‐of‐the‐art direct construction methods from the literature for producing globally D‐optimal designs for virtually any number of two‐level factors as well as any N greater than the number of factors. We summarize the known methods as well as areas for continued research, with the intention of catalyzing research in extending these construction methods.

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The excess of Hadamard matrices and optimal designs

Cited by 21 publications

References 9 publications

Weaving hadamard matrices with maximum excess and classes with small excess

Weaving hadamard matrices with maximum excess and classes with small excess

Large-determinant sign matrices of order 4k+1

Direct construction of globally D‐optimal designs for factors at two levels and main effects models

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