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A class of latin squares derived from finite abelian groups

Anthony B. Evans (2014)

Commentationes Mathematicae Universitatis Carolinae

We consider two classes of latin squares that are prolongations of Cayley tables of finite abelian groups. We will show that all squares in the first of these classes are confirmed bachelor squares, squares that have no orthogonal mate and contain at least one cell though which no transversal passes, while none of the squares in the second class can be included in any set of three mutually orthogonal latin squares.

A uniqueness result for 3 -homogeneous latin trades

Nicholas J. Cavenagh (2006)

Commentationes Mathematicae Universitatis Carolinae

A latin trade is a subset of a latin square which may be replaced with a disjoint mate to obtain a new latin square. A k -homogeneous latin trade is one which intersects each row, each column and each entry of the latin square either 0 or k times. In this paper, we show that a construction given by Cavenagh, Donovan and Drápal for 3 -homogeneous latin trades in fact classifies every minimal 3 -homogeneous latin trade. We in turn classify all 3 -homogeneous latin trades. A corollary is that any 3 -homogeneous...

Algorithm for the complement of orthogonal operations

Iryna V. Fryz (2018)

Commentationes Mathematicae Universitatis Carolinae

G. B. Belyavskaya and G. L. Mullen showed the existence of a complement for a k -tuple of orthogonal n -ary operations, where k < n , to an n -tuple of orthogonal n -ary operations. But they proposed no method for complementing. In this article, we give an algorithm for complementing a k -tuple of orthogonal n -ary operations to an n -tuple of orthogonal n -ary operations and an algorithm for complementing a k -tuple of orthogonal k -ary operations to an n -tuple of orthogonal n -ary operations. Also we find some...

Biembeddings of symmetric configurations and 3-homogeneous Latin trades

Mike J. Grannell, Terry S. Griggs, Martin Knor (2008)

Commentationes Mathematicae Universitatis Carolinae

Using results of Altshuler and Negami, we present a classification of biembeddings of symmetric configurations of triples in the torus or Klein bottle. We also give an alternative proof of the structure of 3-homogeneous Latin trades.

Commutative subloop-free loops

Martin Beaudry, Louis Marchand (2011)

Commentationes Mathematicae Universitatis Carolinae

We describe, in a constructive way, a family of commutative loops of odd order, n 7 , which have no nontrivial subloops and whose multiplication group is isomorphic to the alternating group 𝒜 n .

Construction of the mutually orthogonal extraordinary supersquares

Cristian Ghiu, Iulia Ghiu (2014)

Open Mathematics

Our purpose is to determine the complete set of mutually orthogonal squares of order d, which are not necessary Latin. In this article, we introduce the concept of supersquare of order d, which is defined with the help of its generating subgroup in 𝔽 d × 𝔽 d . We present a method of construction of the mutually orthogonal supersquares. Further, we investigate the orthogonality of extraordinary supersquares, a special family of squares, whose generating subgroups are extraordinary. The extraordinary subgroups...

Construction, properties and applications of finite neofields

Anthony Donald Keedwell (2000)

Commentationes Mathematicae Universitatis Carolinae

We give a short account of the construction and properties of left neofields. Most useful in practice seem to be neofields based on the cyclic group and particularly those having an additional divisibility property, called D-neofields. We shall give examples of applications to the construction of orthogonal latin squares, to the design of tournaments balanced for residual effects and to cryptography.

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