A Census of Small Latin Hypercubes

McKay, Brendan D.; Wanless, Ian M.

doi:10.1137/070693874

Cited by 51 publications

(41 citation statements)

References 14 publications

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“…Although he did not say so, it is simple to use such examples to create noncompletable n×n×· · ·×n×k latin hypercuboids in higher dimensions. Kochol conjectured that all noncompletable latin cuboids are more than half-full, but examples of noncompletable 5 × 5 × 2, 6 × 6 × 2, 7 × 7 × 3, and 8 × 8 × 4 latin cuboids were subsequently given in [10]. Our results will show that Kochol's conjecture fails for all orders except possibly those that are 1 mod 4.…”

Section: Introductionmentioning

confidence: 88%

“…A species (also known as a main class) is an equivalence class of latin squares. We shall use the term "species" for the natural generalization of this well-known notion to latin cubes and cuboids (see also [10]). For cubes, the easiest way to define species is to write the cube as a set of quadruples (i, j, l, s), where s is the symbol in cell (i, j) of layer l. Then a species of order n latin cubes is an orbit under the natural action of the wreath product S n S 4 on these quadruples.…”

Section: Thin Nonextendible Cuboidsmentioning

confidence: 99%

“…The fewest number of extensions was 3932, which was achieved by the following example: 1 2 3 4 5 6 2 1 4 3 6 5 3 5 1 6 4 2 4 6 5 1 2 3 5 3 6 2 1 4 6 4 2 5 3 1 3 5 1 6 2 4 6 2 5 1 4 3 1 4 3 2 5 6 5 1 4 3 6 2 4 6 2 5 3 1 2 3 6 4 1 5. The most number of extensions was 41984, which was achieved by two species: the one shown below, and the one that can be obtained from this one by interchanging the shaded symbols: 1 2 3 4 5 6 2 1 4 3 6 5 3 4 5 6 1 2 4 3 6 5 2 1 5 6 1 2 3 4 6 5 2 1 4 3 2 1 4 3 6 5 1 2 3 4 5 6 4 3 6 5 2 1 3 4 5 6 1 2 6 5 2 1 4 3 5 6 1 2 3 Thus, the thinnest nonextendible latin cuboid of order 6 is a 6× 6 × 3 latin cuboid. An example of such a latin cuboid is given below; it is an extension of the noncompletable 6 × 6 × 2 latin cuboid given in [10]: 1 2 3 4 5 6 2 1 4 3 6 5 3 4 5 6 1 2 4 3 6 5 2 1 5 6 1 2 3 4 6 5 2 1 4 3 2 1 4 3 6 5 1 2 5 6 4 3 4 5 6 2 3 1 3 6 2 1 5 4 6 4 3 5 1 2 5 3 1 4 2 6 3 4 5 6 1 2 4 3 2 1 5 6 6 1 3 5 2 4 5 2 1 4 6 3 2 5 6 3 4 1 1 6 4 2 3 5. Hence, order 6 is the smallest order for which there exists a noncompletable latin cuboid that has strictly fewer layers than any nonextendible latin cuboid. For order 7, the construction in Theorem 7 gives a nonextendible 7 × 7 × 3 latin cuboid, but we do not know if there is one with fewer layers.…”

Section: Thin Nonextendible Cuboidsmentioning

confidence: 99%

“…Of the 31 species of 5 × 5 × 2 latin cuboids, there is only one that is nonextendible; it is the noncompletable example given in [10]. To find the thinnest example of a nonextendible latin cuboid of order 6, we compiled a catalogue of all 601115 species of 6 × 6 × 2 latin cuboids.…”

Section: Thin Nonextendible Cuboidsmentioning

confidence: 99%

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Nonextendible Latin Cuboids

Bryant¹,

Cavenagh²,

Maenhaut³

et al. 2012

SIAM J. Discrete Math.

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Abstract. We show that for all integers m 4 there exists a 2m × 2m × m latin cuboid that cannot be completed to a 2m× 2m× 2m latin cube. We also show that for all even m > 2 there exists a (2m−1) × (2m−1) × (m−1) latin cuboid that cannot be extended to any (2m−1) × (2m−1) × m latin cuboid.Key words. latin cube, latin cuboid, transversal, orthogonal latin squares AMS subject classification. 05B15 DOI. 10.1137/110825534 1. Introduction. There is a celebrated result due to Marshall Hall [6] that every latin rectangle is completable to a latin square. However, the equivalent statement in higher dimensions is not true. The purpose of this paper is to investigate the extent to which it fails.We think of a 3-dimensional array as having layers stacked on top of each other. It also has lines of cells in three directions, obtained from fixing two coordinates and allowing the third to vary. The lines obtained by varying the first, second, and third coordinates will be known respectively as columns, rows, and stacks. The first, second, and third coordinates themselves will be referred to as the indices of the rows, columns, and layers.An n × n × k latin cuboid is a 3-dimensional array containing n different symbols positioned so that every symbol occurs exactly once in each row and column and at most once in each stack. An n × n × n latin cuboid is a latin cube of order n. Every layer of a latin cuboid is a latin square, and we will present our cuboids by displaying the latin squares corresponding to each layer. Each individual layer is composed of a set of n 2 entries, each of which is a triple (r, c, s), where s is the symbol in row r and column c. The layer in which a given entry resides will always be made clear by the context.We say that an n× n× k latin cuboid has order n. It is extendible if it is contained in some n× n× (k + 1) latin cuboid and it is completable if it is contained in some latin cube of order n. Our aim is to investigate how "thin" (that is, how small k can be, relative to n) nonextendible and noncompletable latin cuboids can be. We will refer to an n × n × k latin cuboid as being less than half-full, half-full, or more than half-full

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Section: Introductionmentioning

confidence: 88%

Section: Thin Nonextendible Cuboidsmentioning

confidence: 99%

Section: Thin Nonextendible Cuboidsmentioning

confidence: 99%

Section: Thin Nonextendible Cuboidsmentioning

confidence: 99%

See 2 more Smart Citations

Nonextendible Latin Cuboids

Bryant¹,

Cavenagh²,

Maenhaut³

et al. 2012

SIAM J. Discrete Math.

Self Cite

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“…There are many Latin hypercubes for xed number of points and dimensions. In fact, when the required number of points and the number of dimensions increase, the number of Latin hypercubes increases exponentially [24]. We use the function maximinLHS from the R package lhs to create Latin hypercubes.…”

Section: Latin Hypercube Samplingmentioning

confidence: 99%

An empirical comparison of some experimental designs for the valuation of large variable annuity portfolios

Gan

Valdez

2016

Dependence Modeling

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Variable annuities contain complex guarantees, whose fair market value cannot be calculated in closed form. To value the guarantees, insurance companies rely heavily on Monte Carlo simulation, which is extremely computationally demanding for large portfolios of variable annuity policies. Metamodeling approaches have been proposed to address these computational issues. An important step of metamodeling approaches is the experimental design that selects a small number of representative variable annuity policies for building metamodels. In this paper, we compare empirically several multivariate experimental design methods for the GB2 regression model, which has been recently discovered to be an attractive model to estimate the fair market value of variable annuity guarantees. Among the experimental design methods examined, we found that the data clustering method and the conditional Latin hypercube sampling method produce the most accurate results.

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Classification of Graeco-Latin Cubes

Kokkala

Östergård

2014

J. Combin. Designs

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A Census of Small Latin Hypercubes

Cited by 51 publications

References 14 publications

Nonextendible Latin Cuboids

Nonextendible Latin Cuboids

An empirical comparison of some experimental designs for the valuation of large variable annuity portfolios

Classification of Graeco-Latin Cubes

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