Some New Latin Power Sets Not Based on Groups

Dénes, J.; Owens, P. J.

doi:10.1006/jcta.1998.2911

Cited by 5 publications

(11 citation statements)

References 9 publications

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“…In order to cut down our search space to a feasible size we use the symmetries of Latin power sets studied in Section 2. In the final section we prove a conjecture of Dénes et al from [4] and [5], and disprove a conjecture made by Dénes [1] and two others by Dénes and Owens [6].…”

Section: Theorem 1 Let L Be a Latin Square Based On A Finite Group Gmentioning

confidence: 70%

“…It should be obvious that a permutation applied to the rows of each square in a Latin power set leaves the set essentially unchanged. In contrast, Latin power sets do not survive permutations of the symbols (this fact was noted in [6], and we will provide an example in Section 3). However, to establish the largest cardinality of a Latin power set of order 10 it is essential that we reduce the search space.…”

Section: Properties Of Latin Power Setsmentioning

confidence: 90%

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Answers to Questions by Dénes on Latin Power Sets

Wanless

2001

European Journal of Combinatorics

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The ith power, L i , of a Latin square L is that matrix obtained by replacing each row permutation in L by its ith power. A Latin power set of cardinality m ≥ 2 is a set of Latin squaresWe prove some basic properties of Latin power sets and use them to resolve questions asked by Dénes and his various collaborators.Dénes has used Latin power sets in an attempt to settle a conjecture by Hall and Paige on complete mappings in groups. Dénes suggested three generalisations of the Hall-Paige conjecture. We refute all three with counterexamples.Elsewhere, Dénes et al. unsuccessfully tried to construct three mutually orthogonal Latin squares of order 10 based on a Latin power set. We confirm as a result of an exhaustive computer search that there is no Latin power set of the kind sought. However we do find a set of four mutually orthogonal 9 × 10 Latin rectangles.We also show the non-existence of a 2-fold perfect (10, 9, 1)-Mendelsohn design which was conjectured to exist by Dénes. Finally, we prove a conjecture originally due to Dénes and Keedwell and show that two others of Dénes and Owens are false.

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Section: Theorem 1 Let L Be a Latin Square Based On A Finite Group Gmentioning

confidence: 70%

Section: Properties Of Latin Power Setsmentioning

confidence: 90%

Answers to Questions by Dénes on Latin Power Sets

Wanless

2001

European Journal of Combinatorics

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“…В общем случае введенное произведение латинских квадратов (ква-зигрупп) может не быть латинским квадратом (квазигруппой). В статьях [15][16][17] латинских квадратов порядка n для любого n ≠ 2, 6 и доказано, что эта гипотеза верна для всех n при 0 50 n ≤ ≤ и всех n, бόльших 50, исключая, возможно, числа вида 6k + 2. Их метод был основан на известной схеме Мендельсона [21], которая дает бесконечно много конт-примеров к гипотезе Эйлера, но, к сожалению, не применима при n ≡ 2 (mod 6).…”

Section: (5)unclassified

“…Их метод был основан на известной схеме Мендельсона [21], которая дает бесконечно много конт-примеров к гипотезе Эйлера, но, к сожалению, не применима при n ≡ 2 (mod 6). В работе [17] предложен новый способ конструирования степенных множеств латинских квадратов (квазигрупп) порядка р для любого простого 11 р ≥ . (Из них уже можно получать степенные множества квазигрупп и латинских квадратов составных размеров известным способом, используя прямые разложения алгебр.)…”

Section: (5)unclassified

О Методах Построения Систем Ортогональных Квазигрупп С Использованием Групп

Glukhov¹

2011

Матем. вопр. криптогр.

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Предлагается метод построения систем ортогональных квазигрупп и ортогональных латинских квадратов на основе групп Фробениуса. Пока-зано, что в схему этого метода укладываются многие (но не все) известные методы построения ортогональных латинских квадратов с использованием групп.Ключевые слова: квазигруппа, латинский квадрат, система ортогональ-ных квазигрупп (латинских квадратов), регулярное множество подстановок, регулярный автоморфизм группы On a method of construction of orthogonal quasigroup systems by means of groups M. M. Gluhov Academy of Cryptography of Russian Federation, MoscowAbstract. We suggest a method of construction of orthogonal quasigroups systems and orthogonal Latin squares by means of Frobenius groups. It is proved that this method generalizes many (but not all) existing methods of orthogonal Latin squares construction by means of groups.

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“…In the present paper we develop the method of [7] for bounding the character sum (2) (see e.g. [7], [10] …”

Section: -mentioning

confidence: 99%