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Diagonally Cyclic Latin Squares

Wanless, Ian

Description

A latin square of order n possessing a cyclic automorphism of order n is said to be diagonally cyclic because its entries occur in cyclic order down each broken diagonal. More generally, we consider squares possessing any cyclic automorphism. Such squares will be named after Parker, in recognition of his seminal contribution to the study of orthogonal latin squares. Our primary aim is to survey the multitude of applications of Parker squares and to collect the basic results on them together in...[Show more]

dc.contributor.authorWanless, Ian
dc.date.accessioned2015-12-13T23:14:39Z
dc.identifier.issn0195-6698
dc.identifier.urihttp://hdl.handle.net/1885/88719
dc.description.abstractA latin square of order n possessing a cyclic automorphism of order n is said to be diagonally cyclic because its entries occur in cyclic order down each broken diagonal. More generally, we consider squares possessing any cyclic automorphism. Such squares will be named after Parker, in recognition of his seminal contribution to the study of orthogonal latin squares. Our primary aim is to survey the multitude of applications of Parker squares and to collect the basic results on them together in a single location. We mention connections with orthomorphisms and near-orthomorphisms of the cyclic group as well as with starters, even starters, atomic squares, Knut Vik designs, bachelor squares and pairing squares.In addition to presenting the basic theory we prove a number of original results. The deepest of these concern sets of mutually orthogonal Parker squares and their interpretation in terms of orthogonal arrays. In particular we study the effect of the various transformations of these orthogonal arrays which were introduced by Owens and Preece.Finally, we exhibit a new application for diagonally cyclic squares; namely, the production of subsquare free squares (so called N∞ squares). An explicit construction is given for a latin square of any odd order. The square is conjectured to be N∞ and this has been confirmed up to order 10 000 by computer. This represents the first published construction of an N∞ square for orders 729, 2187 and 6561.
dc.publisherElsevier
dc.sourceEuropean Journal of Combinatorics
dc.titleDiagonally Cyclic Latin Squares
dc.typeJournal article
local.description.notesImported from ARIES
local.description.refereedYes
local.identifier.citationvolume25
dc.date.issued2004
local.identifier.absfor010104 - Combinatorics and Discrete Mathematics (excl. Physical Combinatorics)
local.identifier.ariespublicationMigratedxPub18511
local.type.statusPublished Version
local.contributor.affiliationWanless, Ian, College of Engineering and Computer Science, ANU
local.description.embargo2037-12-31
local.bibliographicCitation.startpage393
local.bibliographicCitation.lastpage413
local.identifier.doi10.1016/j.ejc.2003.09.014
dc.date.updated2015-12-12T08:39:46Z
local.identifier.scopusID2-s2.0-0842325241
CollectionsANU Research Publications

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