On the possible volumes of μ -way latin trades

Adams, Peter; Billington, Elizabeth J.; Bryant, Darryn; Mahmoodian, E. S.

doi:10.1007/s00010-002-8026-4

Cited by 12 publications

(19 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As IS There does not exist a large set of idempotent latin squares of order n = 6, however there does exist a (4,5,6)-latin trade given by: (2,3,4,5) • (1, 4, 5, 3) (5, 2, 1, 4) (3, 5, 2, 1) (4,1,3,2) • (3, 2, 5, 4) (6, 3, 4, 5) (4, 5, 2, 6) (5, 6, 3, 2) (2,4,6,3) (1, 4, 5, 3) (4, 5, 3, 6) (5, 1, 6, 4) • (6, 3, 1, 5) (3, 6, 4, 1) Applying Theorem 7 of [7] to the combination of a (3,5,6) …”

Section: Resultsmentioning

confidence: 99%

“…An extension of this is to consider the µ-way intersections of the structures, and work has been done taking the underlying structure to be Steiner Triple Systems in [73], m-cycle systems in [2], and latin squares in [3] and [1].…”

Section: Introductionmentioning

confidence: 99%

“…After calling Perm.next(), Perm will contain the permutation (0,1,2,3,5,4), and after calling it a second time Perm will contain the permutation (0,1,2,5,3,4).…”

Section: Permutation Objects and Representing Cyclic Mnolsmentioning

confidence: 99%

Section: Permutation Objects and Representing Cyclic Mnolsmentioning

confidence: 99%

“…Suppose the permutation object Perm contains the permutation (2,1,3,4,5,0). This permutation is not nearly orthogonal to the identity permutation (0,1,2,3,4,5) because both contain the symbol 1 in the second entry. If we wish to find the next permutation that is nearly orthogonal to the identity permutation, clearly the next few permutations of (2,1,3,5,0,4), (2,1,3,5,4,0), etc.…”

Section: Permutation Objects and Representing Cyclic Mnolsmentioning

confidence: 99%

See 4 more Smart Citations

Latin Squares and Related Structures

Marbach¹

View full text Add to dashboard Cite

A latin square of order n is an n × n array of cells, each filled with one of n symbols such that each row and each column contain each symbol precisely once. This thesis contributes three new results from three different topics within the study of latin squares.In doing this, we give a broad overview of three distinct areas of the study of latin squares, and the results that have been accomplished in the literature so far.The first study presents new results pertaining to transversals in latin squares. Previous work on transversals has investigated the spectrum of intersection sizes of two transversals within the back-circulant latin squares. A natural extension to this work is to investigate the spectrum of intersection sizes of more than two transversals within the back-circulant latin squares. In this thesis we will accomplish this for three and four transversals, and for all but a finite list of exceptions give a design theoretic construction that recursively builds from base designs that we found by a computational search.The second study investigates µ-way k-homogeneous latin trades. These structures have been extensively studied when µ = 2, but much less is known when µ > 2. Previous investigation had filled in a fraction of the spectrum when µ = 3. We continue this study giving new constructions and show that 3-way k-homogeneous latin trades of order m exist for all but 196 possible exceptions.The third study investigates mutually nearly orthogonal latin squares (MNOLS). These MNOLS are similar to mutually orthogonal latin squares, and can also be used in the design of experiments. Continuing from previous investigations, we enumerate the number of collections of three cyclic MNOLS for latin squares with order up to 16. This required using computational enumeration techniques and a large optimised computer search, as the search space was incredibly large. We present the number of collections of three MNOLS for latin squares with order up to 16 under a variety of equivalences, where we take the collections to be either sets or ordered lists.ii Declaration by authorThis thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis.

show abstract

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…After calling Perm.next(), Perm will contain the permutation (0,1,2,3,5,4), and after calling it a second time Perm will contain the permutation (0,1,2,5,3,4).…”

Section: Permutation Objects and Representing Cyclic Mnolsmentioning

confidence: 99%