1961
DOI: 10.4153/cjm-1961-031-7
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Orthomorphisms of Groups and Orthogonal Latin Squares. I

Abstract: Euler (6) in 1782 first studied orthogonal latin squares. He showed the existence of a pair of orthogonal latin squares for all odd n and conjectured their non-existence for n = 2(2k + 1). MacNeish (8) in 1921 gave a construction of n — 1 mutually orthogonal latin squares for n = p with p prime and of n(v) mutually orthogonal squares of order v wherewith p1 p2, … , Pr being distinct primes and

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Cited by 106 publications
(40 citation statements)
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References 13 publications
(12 reference statements)
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“…If G is an abelian group and θ 2 is orthogonal to θ 1 , then θ 1 is orthogonal to θ 2 too. This follows from the well-known fact that, for an abelian group (G, +), the inverse of the complete mapping (orthomorphism) is also a complete mapping (orthomorphism) ( [5,13] (2.3), then all of its parastrophes /, \, //, \\, · can be also derived by an orthomorphisms (see [5] and [26]). This fact can be especially useful for designing cryptographic primitives, like encoding and decoding functions.…”
Section: Diagonal Methods and Orthomorphismsmentioning
confidence: 93%
“…If G is an abelian group and θ 2 is orthogonal to θ 1 , then θ 1 is orthogonal to θ 2 too. This follows from the well-known fact that, for an abelian group (G, +), the inverse of the complete mapping (orthomorphism) is also a complete mapping (orthomorphism) ( [5,13] (2.3), then all of its parastrophes /, \, //, \\, · can be also derived by an orthomorphisms (see [5] and [26]). This fact can be especially useful for designing cryptographic primitives, like encoding and decoding functions.…”
Section: Diagonal Methods and Orthomorphismsmentioning
confidence: 93%
“…Before we formulate the conditions of Theorem 1 of [1] for T -quasigroups we recall that a permutation α of a group Q(+) is called an orthomorphism (respectively, a complete mapping) if x − αx = βx(x + αx = βx), where β is a permutation of Q and −x = I x is the inverse element for x in the group Q(+) [4].…”
Section: Check Character Systems Over T −Quasigroupsmentioning
confidence: 99%
“…В работе [21] для групп небольших порядков (от 3 до 12) поиск орто-морфизмов группы производится с помощью компьютерных вычислений. Используя ортоморфизмы групп, авторы находят максимальные системы ОЛК небольших порядков.…”
Section: (5)unclassified
“…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). В работе [17] предложен новый способ конструирования степенных множеств латинских квадратов (квазигрупп) порядка р для любого простого 11 р ≥ .…”
Section: (5)unclassified