2-(v，k，1)设计的可解线-传递自同构群的开题报告.docx

2-(v，k，1)设计的可解线-传递自同构群的开题报告

Introduction

Thestudyofintegralandlineardesignsisanimportantfieldofcombinatoricsanddesigntheory.Integraldesignsareaclassofcombinatorialdesignsthatcanbeusedinvariousapplicationssuchaserror-correctingcodes,cryptography,andcommunicationsystems.Lineardesigns,ontheotherhand,areasubclassofintegraldesignsthathaveadditionalpropertiesrelatedtolinearcodesanderror-correctingcapabilities.

Inthisreport,wewillfocusonthestudyof2-(v,k,1)designs,whichareaspecifictypeofintegraldesignwithinterestingproperties.Wewillalsodiscusstheconceptofself-embeddingandself-isomorphism,whichareimportantconceptsindesigntheory.Finally,wewillpresentsomeresultsonthesolvabilityandtransitivityofself-isomorphic2-(v,k,1)designs.

2-(v,k,1)Designs

A2-(v,k,1)designisacollectionofk-elementsubsetsofav-elementsetsuchthateachpairofelementsiscontainedinexactlyonesubset.Inotherwords,a2-(v,k,1)designisasetofmutuallyorthogonalLatinsquaresoforderv.

Theconceptof2-(v,k,1)designsiscloselyrelatedtotheconceptoffinitegeometries,wherepointsarerepresentedbyelementsofasetandlinesarerepresentedbysubsetsofthatset.Infact,everyfiniteprojectivegeometrycanberepresentedasa2-(v,k,1)design.

Oneimportantpropertyof2-(v,k,1)designsisthattheyaresymmetric,meaningthateverytwopointsofthedesignoccurinexactlythesamenumberofblocks.Thispropertyhasimportantconsequencesforthesymmetryoftheautomorphismgroupofthedesign.

Self-embeddingandSelf-isomorphism

AdesignDissaidtobeself-embeddingifthereexistsaninjectivefunctionfromthesetofpointsofDtothesetofblocksofDthatpreservestheincidencerelation.Inotherwords,thisfunctionmapseachpointtoablockthatcontainsit,andeachblocktoasetofpointsthatitcontains.

AdesignDissaidtobeself-isomorphic