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Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2013-04-30.

Permanent Link: http://ufdc.ufl.edu/UFE0042803/00001

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Title: Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2013-04-30.
Physical Description: Book
Language: english
Creator: DUCEY,JOSHUA E
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

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Subjects / Keywords: Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Statement of Responsibility: by JOSHUA E DUCEY.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Sin, Peter.
Electronic Access: INACCESSIBLE UNTIL 2013-04-30

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042803:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042803/00001

Material Information

Title: Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2013-04-30.
Physical Description: Book
Language: english
Creator: DUCEY,JOSHUA E
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Statement of Responsibility: by JOSHUA E DUCEY.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Sin, Peter.
Electronic Access: INACCESSIBLE UNTIL 2013-04-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042803:00001


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PROBLEMSINALGEBRAICCOMBINATORICS By JOSHUAE.DUCEY ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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c r 2011JoshuaE.Ducey 2

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Idedicatethiswork: Tomyfamily,especially: LawrenceandRuthDucey;MelissaandMichaelPoteat;Mallor y,Alex,Julie,andNick Yourunconditionalloveandsupporthavesurelymademestro ngerthanIcanknow. ToMattBranco,myoldestfriend Tomypug,Aro Finally,tomyfuturewife,MinahOh Youhavemadethelastfewyearsofmylifefeellikeagreatadv enture,andIcannotwait tobeginanewadventurewithyou. 3

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ACKNOWLEDGMENTS Imustbeginbythankingmyadvisor,Dr.PeterSin.Youhaveta ughtmemuch aboutmathematics,andalsoaboutbeingamathematician.Th ankyouforourmany discussions,andforshowingmethingsthatIcanthinkabout fortherestofmylife.I consideritagreathonortohavebeenyourstudent. Totheothermembersofmycommittee:Dr.RichardCrew,Dr.Pi erreRamond,Dr. PaulRobinson,andDr.AlexTurull;Ithankyouallforthecom mentsmade,questions asked,andadvicegivenduringthecompletionofthiswork.Y ouhaveeachmademea bettermathematicianinsomeway. IwishtothankDr.Mikl osB onaforteachingmehowtocount,andtothankDr.Yuli Rudyakforteachingmetopology.ToJimDavis,Ithankyoufor instillingcondencein measmyteacher,andforthesupportyouhavegivenmeasmycol league.Ifnotfor you,Iwouldnotbehere. Manynumericalexperimentswereperformedduringthecompl etionofthiswork.I wouldliketothankthecreatorsanddevelopersofSage(www. sagemath.org)formaking availablesuchpowerfulmathematicalsoftware.Thanksals ototheteambehindthe exactlinearalgebrasoftwareLinBox(www.linalg.org),wh ichImademuchuseof. IamgratefultotheUniversityofFloridaMathematicsDepar tmentforbeinga stimulatingresearchenvironment,andalsoforgivingmeth eopportunitytodiscover thatIlovetoteach.IthanktheUniversityofFloridaColleg eofLiberalArtsandSciences forawardingmewiththeKeeneDissertationFellowship,whi chhelpedtomakemynal semesterhereaveryproductiveone.Forthegeneroussuppor tIreceivedthroughthe ChatYinHoMemorialScholarship,Isincerelythankthefami lyandfriendsofProfessor Ho. Themostimportantresultsofthisworkmakeupapaperco–aut horedbyDr. AndriesE.Brouwer,Dr.PeterSin,andmyself,thatiscurren tlybeingrefereed.Iam thankfulfortheuniqueperspectivesandtalentsofthesetw o,andIamconvincedthat 4

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theratherdifferentwaysinwhichweeachapproachedthepro blemwerecriticaltoour success.Finally,IthanktheBanffInternationalResearch Station,wherediscussionof thisworkbeganataworkshopinMarchof2009. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1Introduction ...................................... 10 1.1IncidenceMatrices ............................... 10 1.2StatementoftheProblem ........................... 11 1.3SomeMotivation ................................ 11 2PreliminariesandDenitions ............................ 14 2.1TheSmithNormalForm ............................ 14 2.2Localization ................................... 16 2.3IncidenceMapsandRepresentationTheory ................. 17 2.4GraphTheoryandMatrixIdentities ...................... 18 3TheMainResult ................................... 21 3.1StatementsofTheorems ............................ 21 3.2ExamplesandCalculations .......................... 24 4ElementaryDivisors ................................. 27 4.1TheModules M i and N j ............................ 27 4.2SmithNormalFormBases .......................... 29 5ProofsofTheorems ................................. 33 5.1ProofofTheorem3.1 ............................. 33 5.2TheGeneralResult .............................. 36 APPENDIX:SAMPLESAGEPROGRAM ........................ 46 REFERENCES ....................................... 47 BIOGRAPHICALSKETCH ................................ 49 6

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LISTOFTABLES Table page 3-1LinBoxcomputationsforsomesmallvaluesof q = p t .............. 25 3-2Thecoefcients d i thatarisewhencalculating d ( ~ s ) inTheorem3.2. ...... 25 3-3TheSmithnormalformoftheincidencematrixofskewline sin PG(3, p ) .... 26 4-1Visualizingthe R -submodules M i ( ) ........................ 30 4-2Visualizingthe R -submodules N j ( ) ........................ 31 7

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LISTOFFIGURES Figure page 1-1Theincidencematrixofskewlinesin PG(3,4) .................. 13 5-1IllustratingLemma5.2when n =3 and r = s = t =2 .............. 42 8

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy PROBLEMSINALGEBRAICCOMBINATORICS By JoshuaE.Ducey May2011 Chair:PeterSinMajor:Mathematics ThemainresultofthisworkisthedeterminationoftheSmith normalformofthe incidencematrixoflinesvs.linesin PG(3, q ) ,where q = p t isaprimepowerand twolinesaredenedtobeincidentifandonlyiftheyareskew .Thisprincipalresultis essentiallyacorollaryofamoregeneraltheorem.Inordert oprovethegeneraltheorem, wedevelopsomenewideasinthebasictheoryofelementarydi visors,andalsoemploy somerepresentationtheory.Asanothercorollarytothegen eraltheorem,weobtain somespecicknowledgeofthe p -adicelementarydivisorsoftheincidencematrixof r -dimensionalsubspacesvs. s -dimensionalsubspacesin PG( n q ) ,whereincidence againmeanszero–intersection. 9

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CHAPTER1 INTRODUCTION 1.1IncidenceMatrices Oneofthemostubiquitousconceptsinmathematicsisthatof anincidence structure.Thisisjustatriple ( X Y I ) consistingoftwosetsofobjectstogetherwith arelation I X Y betweenthem.Ifanobjectintherstsetisrelatedtoanobje ctin thesecondset,wesaythatthesetwoare“incident.”Oftenth esetsarerequiredtobe disjoint,butthatisnotimportanttous.Wewillhoweverdea lexclusivelywithnitesets. Givenaniteincidencestructure,itcanbeencodedintoare ctangulararrayas follows.Lettherowsofthearraycorrespondtotherstseto fobjects;thecolumns correspondtothesecondsetofobjects.Thenweplaceaonein the ( i j ) -positionof thearrayiftheobjectcorrespondingtorow i isincidentwiththeobjectcorresponding tocolumn j ,otherwisethatpositiongetsazero.Suchanarrayiscalled an incidence matrix Sincetheincidencematrixcarriesalloftheinformation,i tisagoodthingto study.Variousnumericalinvariantsofthematrixnowbecom einvariantsofthe incidencestructure.Veryoftenthesematricesarisefromg eometricorcombinatorial considerations,soinasensetheseinvariantsareanalogou stothehomologyorEuler characteristicofatopologicalspace. Consider,forexample,thesituationwhenthematrixissqua re(sobothsetshave thesamesize).Iffurthermorethetwosetsareequalandther elationissymmetric,then theincidencestructureisjustagraph,andthematrixisusu allycalledanadjacency matrix(seeSection 2.4 ).Naturalinvariantstoconsideraretheeigenvalues,andc ertain propertiesofthegrapharereectedinthespectrumofthema trix. Fornon–squarematrices,therankoftheincidencematrixis agoodchoiceof invariant.Bychangingtheeldthatyouviewthematrixentr iestobecomingfrom,the rankmaychange.Therankoveraeldofcharacteristic p isusuallycalledthe p -rank 10

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ofthematrix.Anincidencematrixisinparticularanintege rmatrix,sowecanalso consideritsSmithnormalform.Thisisjustsomeuniquelyde termineddiagonalmatrix, seeSections 2.1 and 2.2 fordetails.TheSmithnormalformisaratherstronginvaria nt, inthesensethatfromitonecanimmediatelydeducetheranka nd p -rankofthematrix, foranyprime p .ForinformationabouttherelationshipbetweentheSmithn ormalformof anintegermatrixanditsspectrum,see[ 10 11 ]. 1.2StatementoftheProblem Let V bea 4 -dimensionalvectorspaceovertheniteeld F q of q = p t elements, where p isaprime.Wedeclaretwo 2 -dimensionalsubspaces U and W tobeincidentif andonlyif U \ W = f 0 g .Orderingthe 2 -dimensionalsubspacesinsomearbitrarybut xedmanner,wecanformtheincidencematrix A ofthisrelation.Thegoalistocompute theSmithnormalformof A asanintegermatrix. 1.3SomeMotivation Itisusefultoviewthisprobleminamoregeneralcontext.Su pposenowthat V is an ( n +1) -dimensionalvectorspaceovertheniteeld F q of q = p t elements.Denote by L r thesetof r -dimensionalsubspacesof V .So L 1 denotesthepoints, L 2 denotes thelines,etc.oftheprojectivegeometry P ( V ) .Denean r -dimensionalsubspace U and an s -dimensionalsubspace W tobeincidentifandonlyif U \ W = f 0 g ,anddenoteby A r s the jL r jjL s j incidencematrix. Thesematrices A r s arenaturallyinteresting,andmathematicianshavebeen studyingthemsinceatleastthe 1960 s.Thereaderisreferredtothesurveys[ 16 17 ] (seealso[ 4 ,Introduction]).Bysetting n =3 r = s =2 ,and A = A 2,2 ,werecoverthe situationdescribedintheaboveproblem.Bythewell–known Kleincorrespondence[ 7 Chapter15](identifyingthelinesin PG(3, q ) withthepointsofahyberbolicquadricin PG(5, q ) ), A mayalsoberegardedastheadjacencymatrixofthenon–colli nearitygraph onthepointsoftheKleinquadric.Ingeneral,when r = s thesematricescanbeviewed asadjacencymatricesof q -analoguesoftheKnesergraphs. 11

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When r =1 ,theincidencestructureisthatofa 2 -designwith“classicalparameters,” andtheseincidencematricesarethegeneratormatricesofc odescloselyrelatedto theReed–Mullercodes[ 1 ].Thisiswhatinitiallymotivatedtheirstudy,andinthisc ase theirSmithnormalformshavebeenfound[ 4 9 12 ].The p -rankof A r s hasbeenfound ingeneral[ 13 ],butwhenneither r nor s isoneitsSmithnormalformisnotknown.It isthusnaturaltoconsiderwhenboth r and s aregreaterthanone,andtheproblem describedaboveisjusttherstnontrivialcase. Onecanchoosetoconsidernotionsofincidenceotherthanze ro–intersection. Abasicexampleisthesubspace–inclusionrelation;thatis ,twosubspaceswould becalledincidentifthesmalleronewerecontainedinthela rger.Theseincidence structuresareobvious q -generalizationsofcorrespondingrelationsbetweensubs ets ofaniteset.Observe,however,thatasubset T iscontainedinanothersubset K if andonlyif T isdisjointfromthecomplementof K .Thuswhendealingwith sets the inclusionandempty–intersectionrelationsarereallythe samething,andinthiscase theintegerinvariantshavebeenfound[ 15 ].Thisdoesnotcarryovertospaces.The subspace–inclusionrelationismuchmoredifculttounder standthanzero–intersection: itisstillanopenproblemtocalculatethe p -ranksoftheinclusionmatrices. 12

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Figure1-1.Theincidencematrixoflinesvs.linesin PG(3,4) ,wheretwolinesare incidentwhenskew.Ablackpixelisa 1 ,awhitepixelisa 0 13

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CHAPTER2 PRELIMINARIESANDDEFINITIONS 2.1TheSmithNormalForm Asquarematrixwithintegerentriesiscalled unimodular ifitsdeterminantis 1 .If M and N are m n matriceswithintegerentries,thenwesay M and N are equivalent if thereexistunimodularintegermatrices P and Q with PMQ 1 = N Thisisanequivalencerelationonthesetof m n integermatrices.The Smithnormal form ofanintegermatrix M isjustaparticulardiagonalmatrixthatrepresentsthecla ss of M .Precisely,if M isan m n integermatrix,thenthereexistunimodularinteger matrices P and Q suchthatthematrix S ( M )= PMQ 1 =( d i j ) satises d i j =0, for i 6 = j and d i i divides d i +1, i +1 for 1 i < min f m n g Thisdivisibilityconditiondetermines S ( M ) uptothesignofthediagonalentries,and itisalwayswiththisunderstandingthatwereferto S ( M ) as“the”Smithnormalform of M .Thenonzerodiagonalentriesof S ( M ) ,countedwithmultiplicity,arecalledthe invariantfactors ofthematrix M .Breakingaparttheinvariantfactorsintopowersof distinctprimes,wegetthe elementarydivisors of M .Thesearealsodeterminedupto sign,andcountedwithmultiplicity.Acoupleexamplesshou ldmakeallofthisclear. Example2.1. Let M = 0B@ 123456 1CA .Choosing P = 0B@ 104 1 1CA Q 1 = 0BBBB@ 1 21 01 2 001 1CCCCA 14

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wehave S ( M )= PMQ 1 = 0B@ 100030 1CA Theinvariantfactorsare 1 and 3 ,eachoccurringwithmultiplicityone.Theelementary divisorsarealso 1 and 3 ,eachoccurringwithmultiplicityone. Example2.2. Let M = 0BBBB@ 31672522 263212138 582496 4842232 34224 5642844 13116 1CCCCA Setting P = 0BBBB@ 3135212 25 1 1CCCCA Q 1 = 0BBBBBBB@ 3232 64 4 5 5 50 2 7485 1CCCCCCCA wehave S ( M )= PMQ 1 = 0BBBB@ 2000012000000 1CCCCA Theinvariantfactorsare 2 and 12 ,eachoccurringwithmultiplicityone.Theelementary divisorsare 2 4 ,and 3 ,eachoccurringwithmultiplicityone. Manyveryinterestingexamplescanbefoundin[ 11 ].Sinceknowledgeofthe invariantfactorsisequivalenttoknowledgeoftheelement arydivisors,wewillbe focusingourattentiononthelatter.Ourterminologyisrea sonablystandard,although intheliteratureonendsanapparentdisputeoverwhatthee xactmeaningofthese termsshouldbe.TheSmithnormalformisnamedafterHenryJo hnStephenSmith (1826–1883).Asidefrombeingaverytalentedscholarandad ministrator,allaccounts ofhislifeseemtodescribehimasacharmingandmodestmanwh onevermadean enemyorlostafriend[ 5 ]. 15

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2.2Localization Sinceincidencematricesarezero–onematrices,wecanview theirentriesas comingfromanycommutativering R .When R isaprincipalidealdomain,thereis acompletelyanalogousnotionofSmithnormalform“over R .”Seeforexample[ 10 ]. Indeed,thestatementthatanintegermatrixhasaSmithnorm alformisreallyjust amatrix–theoreticdescriptionofthestructuretheoremfo rnitelygeneratedabelian groups,andthistheoremgeneralizestonitelygeneratedm odulesoverprincipalideal domains[ 8 ,Chapters4,7].Inthiscontextofmatricesover R ,thestatementthat P and Q areunimodularmeansthattheyhaveentriescomingfrom R andtheirdeterminants areunitsinthisring(thepurposeofthisconditionistogua ranteethat P 1 and Q 1 also haveentriesin R ).ThediagonalentriesoftheSmithnormalformover R areunique uptomultiplicationbyaunitin R ,andwhenspeakingofinvariantfactorsorelementary divisorsoverthisringwedonotdistinguishbetweenassoci ates.Thuswhenwespeak ofthemultiplicityofaparticularinvariantfactororelem entarydivisor,wearereally countingthenumberofoccurrencesofitsassociates. Animportantspecialcaseiswhen R isanextensionringoftheringofintegers Z Noticethatif P and Q areunimodularasintegermatricesthentheyarestillunimo dular asmatricesover R .Thusif M isanintegermatrixand S ( M ) isitsSmithnormalform over Z ,then S ( M ) istheSmithnormalformof M over R .However,iftherearenon–unit elementsof Z thatbecomeunitsinthering R ,thentheelementarydivisormultiplicities overeachringneednotbethesame.Moregenerally,similars tatementsholdtruewhen R containsahomomorphicimageof Z Example2.3. Take R = Q tobetherationalnumbersand M tobethematrixfrom Example 2.2 .Thenthenonzerodiagonalentriesof S ( M ) areallunitsin Q ,sothat(as arationalmatrix) M has 1 asitsonlyelementarydivisor,occurringwithmultiplicit ytwo. Observethattherankofanintegermatrixisjustthenumbero fnonzerodiagonalentries ofitsSmithnormalformover Z 16

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Example2.4. Take R = F 3 = Z = 3 Z tobetheniteeldof 3 elements,andagainlet M bethematrixfromExample 2.2 .Let M bethematrixobtainedbyreducingallentriesof M (mod3) .Then 1 istheonlyelementarydivisorof M ,occurringwithmultiplicityone. Ingeneral,the p -rankofanintegermatrixisjustthenumberofdiagonalentr iesofits Smithnormalformover Z thatarenotdivisibleby p Example2.5. Take R = Z p tobetheringof p -adicintegers, M thematrixfrom Example 2.2 .Theelementsof Z thatbecomeunitsinthering Z p arepreciselythe integersnotdivisibleby p .Thusasamatrixover Z 2 ,theelementarydivisorsof M are 2 and 4 bothwithmultiplicityone.Over Z 3 ,theelementarydivisorsof M are 1 and 3 both withmultiplicityone.Foranyother(rational)prime p M asamatrixover Z p has 1 asits onlyelementarydivisor,occurringwithmultiplicitytwo. AlloftheseexamplesshowthattheSmithnormalformofanint egermatrixcarriesa greatdealofinformation.Thelastexampleinparticularsh owsthatifweareinterested onlyinthe p -elementarydivisorsofanintegermatrix(thatis,thoseel ementarydivisors thatarepositivepowersofaparticularprime p ),thenwemaychoosetoviewthematrix entriesascomingfrom Z p ratherthan Z .Wewillalwaysbeclearaboutwhichringwe considerourmatrixentriestobecomingfrom. 2.3IncidenceMapsandRepresentationTheory Representationtheorycanbeaverypowerfultoolinthestud yofincidence matrices.Thisisbecausetheincidencestructuresthatare mostinterestingusually havesomegroupactingonthem.Tobeclear,consideranincid encestructure ( X Y I ) where X and Y arenitesetsand I istheincidencerelation I X Y Orderingthesets X and Y ,weformtheincidencematrix M ofthisrelation.Whenwe viewthematrixentriesascomingfromsomecommutativering R ,the j X jj Y j matrix M 17

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representsahomomorphismoffree R -modules : R X R Y where R Z consistsofall R -valuedfunctionsontheset Z .Sincewewillnotbeneeding this“functionnotation,”weidentifyeachelementof Z withitscharacteristicfunction,and view R Z asconsistingofformal R -linearcombinationsoftheelementsof Z .Withthese identications,theabovemapping sendseachelementof x 2 X tothesumofthe elementsof Y thatareincidentwith x ,andbylinearitythispropertycompletelydenes themap.Ifthering R isaeld,thentherankof M isjustthedimensionoftheimage of .For R aprincipalidealdomain,thereismodule–theoreticdescri ptionoftheSmith normalformof M (seeChapter 4 ). Nowif G isanitegroupactingtransitivelyonthesets X and Y ,then R X and R Y arepermutationmodulesforthegroupring RG .Furthermore,iftheactionof G preservestheincidencerelation,i.e. ( x y ) 2I implies ( gx gy ) 2I forall g 2 G then becomesahomomorphismof RG -modules.Thustheimage,kernel,etc.of are RG -submodules,andingeneralthisplacesextremelysevereli mitationsonwhat theirstructurecanbe.Inturn,thisinformationbecomesre levantandusefultoourstudy oftheincidencestructure'sinvariants.Ascanbeexpected ,variousdifcultiesarise andtechniquesareuseddependingonthering R .Mostofwhatweneedcanbefound nicelysummarizedin[ 9 ,AppendicesD,E,F]. 2.4GraphTheoryandMatrixIdentities A graph ( V E ) isaset V of vertices andacollection E of 2 -elementsubsetsof V called edges .If f x y g2E ,thenthevertices x and y aresaidtobeadjacent.Graph theoryterminologycanvaryconsiderably,andwhatweareca llingagraphsomeauthors wouldcalla“simple”or“loopless”graph. 18

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Alternatively,forusagraphisjustanincidencestructure ( V V I ) where I isa symmetricrelationandnoelementof V isincidentwithitself.Orderingtheset V ,we canformtheincidencematrixofthisrelation.Thissymmetr icmatrixisusuallycalledthe adjacencymatrix ofthegraph. Agraphiscalled regular withvalency k ifeachvertexisadjacenttoexactly k other vertices.Agraph ( V E ) is stronglyregular withparameters v k if 1. jVj = v 2. thegraphisregularofvalency k 3. if x and y areadjacentvertices,thenthereareexactly verticesadjacenttoboth x and y 4. if x and y are(distinct)non–adjacentvertices,thenthereareexact ly vertices adjacenttoboth x and y If M istheadjacencymatrixofastronglyregulargraphwithpara meters v k then M satisestheequation M 2 = kI + M + ( J M I ), (2–1) where I and J respectivelydenotetheidentitymatrixandall–onematrix ofthesame sizeas M .Thereasonthatthisequationholdsfollowsfromamoregene ralfact explainedbelow.Fromthisequationitisnotdifculttoded uceboththeeigenvalues of M andtheirmultiplicities,wereferthereaderto[ 3 ]. ToseewhyEquation( 2–1 )holds,considermoregenerallythefollowingsituation. Let ( X Y I ) and ( Y Z J ) betwoniteincidencestructures,andxanorderingof X Y ,and Z .Withrespecttotheseorderings,formthe j X jj Y j incidencematrix M ofthe rstrelation,andthe j Y jj Z j incidencematrix N ofthesecondrelation.Noticethatthe matrixproduct MN hasrowsindexedby X andcolumnsindexedby Z .Let x 2 X and z 2 Z .Withalittlethoughtoneseesthatthe ( x z ) -entryofthematrixproduct MN is 19

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preciselythenumberof y 2 Y thatareincidentwithboth x and z .Insymbols, ( MN ) x z = jf y 2 Y j ( x y ) 2I and ( y z ) 2Jgj Wewillfrequentlymakeuseofthisfact.ThusweseethatEqua tion( 2–1 )isjust expressingproperties 2 3 ,and 4 inthedenitionofastronglyregulargraph. 20

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CHAPTER3 THEMAINRESULT 3.1StatementsofTheorems Wereturnnowtotheproblemdescribedintheintroduction(S ection 1.2 ).Thus V is a 4 -dimensionalvectorspaceovertheniteeld F q of q = p t elements, A = A 2,2 isthe incidencematrixwithrowsandcolumnsindexedbythe 2 -dimensionalsubspacesof V andincidenceisdenedtomeanzero–intersection.Wewillc omputetheSmithnormal formof A asanintegermatrix. Itturnsoutthattheelementarydivisorsof A areallpowersoftheprime p .Aquick waytoseethisistoregard A astheadjacencymatrixofthegraphwithvertexset L 2 ,wheretwolinesareadjacentwhenskew.Thisisastronglyre gulargraph(see Section 2.4 ),withparameters v = q 4 + q 3 +2 q 2 + q +1, k = q 4 = q 4 q 3 q 2 + q = q 4 q 3 Thus A satisestheequation A 2 = q 4 I +( q 4 q 3 q 2 + q ) A +( q 4 q 3 )( J A I ), (3–1) where I and J denotetheidentitymatrixandall–onematrix,respectivel y,ofthe appropriatesizes.Fromthisequationonededucesthatthee igenvaluesof A are q q 2 ,and q 4 withrespectivemultiplicities q 4 + q 2 q 3 + q 2 + q ,and 1 .Since(uptosign, ofcourse) det( A ) istheproductoftheelementarydivisors,weseethattheele mentary divisorsof A areallpowersoftheprime p ThereforedescribingtheSmithnormalformof A amountstospecifyingthenumber oftimeseachprimepower p i i 0 ,occursasanelementarydivisorof A .Thisnumber wedenoteby e i (or e i ( A ) ,whenwewishtoemphasizethematrixunderdiscussion). Sincetheproblemisparametrizedbyourchoiceof q = p t ,itistobeexpectedthatthe elementarydivisormultiplicities e i willdependinsomewayontheexponent t 21

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TheelementarydivisormultiplicitiesarelistedinTable 3-1 ,forsomesmallvaluesof q = p t .Itisworthwhiletoexaminethistablebrieybeforemoving on.Thenexttheorem describessomerelationsthatholdamongthemultiplicitie singeneral,andstudyingthe tableforamomentshouldmakethestatementofthetheoremmu chclearer. Forexample,lookattherowofTable 3-1 correspondingtowhen q =8=2 3 .Pay specialattentiontothefactthathere t =3 .Inthisrowonesees 4(= t +1) nonzero entries,followedby 2(= t 1) zeros,followedby 4(= t +1) nonzeroentries,followed by 2(= t 1) zeros.Thenextentryis 1 ,correspondingtothemultiplicity e 12 (= e 4 t ) ,and allremainingmultiplicitiesarezero.Observethepartial “reversesymmetry”inthetwo chunksofnonzeroentries: e 0 = e 9 (= e 3 t ), e 1 = e 8 (= e 3 t 1 ), e 2 = e 7 (= e 3 t 2 ). (3–2) Addingthemultiplicitiesinthesenonzerochunks,weget e 0 + e 1 + e 2 + e 3 =4160(= q 4 + q 2 ) (3–3) and e 6 + e 7 + e 8 + e 9 =584(= q 3 + q 2 + q ). (3–4) Theorem3.1. Let e i = e i ( A ) denotethemultiplicityof p i asanelementarydivisorof A 1. e 4 t =1 2. e i =0 for t < i < 2 t 3 t < i < 4 t ,and i > 4 t 3. e i = e 3 t i for 0 i < t 4. P ti =0 e i = q 4 + q 2 5. P 3 t i =2 t e i = q 3 + q 2 + q Fromtheidentitiesstatedintheabovetheorem,wecandeduc ealloftheelementary divisormultiplicitiesonceweknow t ofthenumbers e 0 ,..., e t (or t ofthenumbers e 2 t ,..., e 3 t ).Forexample,consideragaintherowofTable 3-1 correspondingto 22

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q =8=2 3 (so t =3 ).Supposeweknowonlythat e 6 =128 e 8 =144 ,and e 9 =216 Thenpart( 5 )ofthetheoremisjustEquation( 3–4 ),andfromthiswecalculate e 7 =96 Inthiscasepart( 3 )becomestheequations( 3–2 ),andwecompute e 0 =216 e 1 =144 and e 2 =96 .FromEquation( 3–3 )(part( 4 )ofthetheorem)wethenget e 3 =3704 Finally, e 12 =1 andallothermultiplicitiesarezerobyparts( 1 )and( 2 ),respectively. Thenexttheoremshowshowtodirectlycomputeeachofthemul tiplicities e 2 t ,..., e 3 t .Bythediscussionabove,thisdataismorethansufcientto determine theSmithnormalformof A .Tostatethetheorem,weneedsomenotation. Set [3] t = f ( s 0 ,..., s t 1 ) j s i 2f 1,2,3 g forall i g and H ( i )= ( s 0 ,..., s t 1 ) 2 [3] t # f j j s j =2 g = i Inotherwords, H ( i ) consistsofthetuplesin [3] t withexactly i twos.Toeachtuple ~ s 2 [3] t weassociateanumber d ( ~ s ) asfollows.For ~ s =( s 0 ,..., s t 1 ) 2 [3] t denethe integertuple ~ =( 0 ,..., t 1 ) by i = ps i +1 s i withthesubscriptsreadmodulo t .Foraninteger k ,set d k tobethecoefcientof x k in theexpansionof (1+ x + + x p 1 ) 4 .Finally,set d ( ~ s )= Q t 1 i =0 d i Theorem3.2. Let e i = e i ( A ) denotethemultiplicityof p i asanelementarydivisorof A Then,for 0 i t e 2 t + i = X ~ s 2H ( i ) d ( ~ s ). Remark. When p =2 ,noticethat d ( ~ s )=0 foranytuple ~ s containinganadjacent 1 and 3 (coordinatesreadcircularly).ThusthesuminTheorem 3.2 issignicantlyeasierto computeinthiscase. 23

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3.2ExamplesandCalculations Toillustratehowtousethesetwotheorems,let'sconsidera nexample. Example3.1. Suppose q =9=3 2 (thatis, p =3 and t =2 ).Thenwehave (1+ x + x 2 ) 4 =1+4 x +10 x 2 +16 x 3 +19 x 4 +16 x 5 +10 x 6 +4 x 7 + x 8 H (0)= f (11),(13),(31),(33) g H (1)= f (21),(23),(12),(32) g H (2)= f (22) g UsingTheorem 3.2 wecompute e 4 = d (11)+ d (13)+ d (31)+ d (33) = d 2 d 2 + d 8 d 0 + d 0 d 8 + d 6 d 6 =10 10+1 1+1 1+10 10 =202, e 5 = d (21)+ d (23)+ d (12)+ d (32) = d 1 d 5 + d 7 d 3 + d 5 d 1 + d 3 d 7 =4 16+4 16+16 4+16 4 =256, e 6 = d (22)= d 4 d 4 =19 19=361. TheremainingnonzeromultiplicitiesarenowgivenbyTheor em 3.1 .Observethatour calculationagreeswithTable 3-1 Weonlyneedafewofthecoefcientsof (1+ x + + x p 1 ) 4 whencomputing d ( ~ s ) Infact,itisnotdifculttocomputethecoefcientsthatwe needexplicitly.Theseare listedinTable 3-2 .Usingthese,wecanincertaincaseswriteclosed–formexpr essions 24

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fortheelementarydivisormultiplicities.Forexample,th ecasewhen q = p (thatis, t =1 )isshowninTable 3-3 Table3-1.LinBoxcomputationsforsomesmallvaluesof q = p t e 0 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 e 11 e 12 q =2 614861 q =3 197120191 q =5 8556570851 q =7 23122191682311 q =2 2 36162203216361 q =3 2 36125660252022563611 q =2 3 216144963704128961442161 Here e i denotesthemultiplicityof p i asanelementarydivisorof A .Anemptyentryinthe tabledenotesa 0 Table3-2.Thecoefcients d i thatarisewhencalculating d ( ~ s ) inTheorem 3.2 (..., s i s i +1 ,...) i d i (...,1,1,...) p 1 p ( p +1)( p +2) = 6= p +2 3 (...,1,2,...)2 p 12( p 1) p ( p +1) = 3=4 p +1 3 (...,2,1,...) p 2( p 1) p ( p +1) = 6= p +1 3 (...,2,2,...)2 p 2 p (2 p 2 +1) = 3=4 p +1 3 + p (...,1,3,...)3 p 1( p 2)( p 1) p = 6= p3 (...,3,1,...) p 3( p 2)( p 1) p = 6= p3 (...,2,3,...)3 p 2( p 1) p ( p +1) = 6= p +1 3 (...,3,2,...)2 p 32( p 1) p ( p +1) = 3=4 p +1 3 (...,3,3,...)3 p 3 p ( p +1)( p +2) = 6= p +2 3 25

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Table3-3.TheSmithnormalformoftheincidencematrixofsk ewlinesin PG(3, p ) ElementaryDivisorMultiplicity 1 p (2 p 2 +1) = 3 pp (3 p 3 2 p 2 +3 p 1) = 3 p 2 p ( p +1)( p +2) = 3 p 3 p (2 p 2 +1) = 3 p 4 1 26

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CHAPTER4 ELEMENTARYDIVISORS 4.1TheModules M i and N j Inthissectionwecollectafewusefulresultsregardingele mentarydivisors. Throughoutthischapterwewillbeworkingoveradiscreteva luationring R .Inother words, R isaprincipalidealdomainwithexactlyonenonzeroprimeid eal.Let p 2 R be aprimegeneratingthisideal.Thisisnotsuchaspecialsitu ation. Example4.1. Let M beanintegermatrix,and p 2 Z aprimeinteger.Theringof integers Z isnotadiscretevaluationring.However,both Z P (thelocalizationof Z atthe primeideal P generatedby p )andthe p -adicintegers Z p arediscretevaluationrings thatcontainacopyof Z .Moreover,for i > 0 themultiplicityof p i asanelementary divisorof M isthesamewhetherweviewtheentriesof M ascomingfrom Z Z P ,or Z p (themultiplicityof p 0 ,i.e.thenumberofelementarydivisorsthatareunits,will ingeneral bedifferentoverdifferentrings). An m n matrixwithentriesin R canbeviewedasahomomorphismoffree R -modulesofniterank: : R m R n Theelementarydivisorsof arebydenitionjusttheelementarydivisorsofthe matrix,andforthexedprime p wealwayslet e i ( ) denotethemultiplicityof p i as anelementarydivisorof Set F = R = pR .If L isan R -submoduleofafree R -module R ` ,then L =( L + pR ` ) = pR ` isan F -vectorspace.For i 0 ,dene M i ( )= f x 2 R m j ( x ) 2 p i R n g and N i ( )= f p i ( x ) j x 2 M i ( ) g 27

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Forconveniencewealsodene N 1 ( )= f 0 g .Thenwehavechainsof R -modules R m = M 0 ( ) M 1 ( ) N 0 ( ) N 1 ( ) andchainsof F -vectorspaces F m = M 0 ( ) M 1 ( ) N 0 ( ) N 1 ( ) Lemma4.1. Let : R m R n beahomomorphismoffree R -modulesofniterank,and let e i ( ) denotethemultiplicityof p i asanelementarydivisorof .Then,for i 0 e i ( )=dim F M i ( ) = M i +1 ( ) =dim F N i ( ) = N i 1 ( ) Proof. Thelemmaiscertainlytrueif =0 ,soassumethat isnonzero.Thenthere isauniquelargestnonnegativeinteger ` with e ` ( ) 6 =0 ;inotherwords, ` isthelargest exponentoccurringamongthepowersof p intheSmithnormalformof .Fromthe theoryofmodulesoverprincipalidealdomains,thereexist sabasis B of R m andabasis C of R n withrespecttowhichthematrixof isinSmithnormalform.Byconsidering thismatrixweareleadtoapartitionof B and C asfollows.For 0 i ` ,let B i bethe elementsof B whoseimageisexactlydivisibleby p i .Ifwelet B ` +1 denotetheelements of B thataremappedtozero,thenwehavethedisjointunion B = ` +1 [ i =0 B i For 0 i ` ,set C i = p i ( B i ) (if B i isempty,justset C i tobetheemptysetalso).Then wehavethedisjointunion C = ` +1 [ i =0 C i 28

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where C ` +1 isdenedtobe Cn[ `i =0 C i .Itiseasytoseethateachofthe R -submodules M i ( ) (resp. N j ( ) )haveabasisconsistingof p -powermultiplesofelementsof B (resp. C ).ThisisdescribedinTable 4-1 andTable 4-2 .Whenwerepresentthemodulesinthis way,thelemmabecomesclear. Table 4-1 andTable 4-2 provideaveryusefulwaytovisualizethe R -modules M i ( ) and N j ( ) ,andreallymakecleartheirconnectionwiththeSmithnorma lformof .The advantagesofthemodule–theoreticapproachtotheSmithno rmalformwillbecome increasinglyapparent.Thisnextresultisveryeasy,butve ryuseful. Lemma4.2. Let r : R m R n beanother R -modulehomomorphism,andsupposethat forsome k 1 wehave ( x ) r ( x )(mod p k ), forall x 2 R m Then e i ( )= e i ( r ), for 0 i k 1. Proof. Verifythat M i ( )= M i ( r ) ,for 0 i k .Theconclusionisnowimmediatefrom Lemma 4.1 4.2SmithNormalFormBases Foragivenhomomorphism : R m R n ,wewillbeinterestedinpairsofbases( B C )withrespecttowhichthematrixof isdiagonal.Wedenea leftSNFbasis for to beanybasis B of R m thatbelongstosuchapair.Similarly,a rightSNFbasis for isany basis C of R n belongingtosuchapair.Wenowdescribehowtoconstructsuc hbases. Suppose : R m R n isnonzero.Thenthereisauniquelargestnonnegative integer ` with e ` ( ) 6 =0 .Wehave M 0 ( ) M 1 ( ) M ` ( ) ) ker( ). 29

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Table4-1.Visualizingthe R -submodules M i ( ) Submodule Basis ( R m =) M 0 = n B 0 B 1 B 2 B 3 B ` 1 B ` B ` +1 M 1 = n p B 0 B 1 B 2 B 3 B ` 1 B ` B ` +1 M 2 = n p 2 B 0 p B 1 B 2 B 3 B ` 1 B ` B ` +1 M 3 = n p 3 B 0 p 2 B 1 p B 2 B 3 B ` 1 B ` B ` +1 ... M ` 1 = n p ` 1 B 0 p ` 2 B 1 p ` 3 B 2 p ` 4 B 3 B ` 1 B ` B ` +1 M ` = n p ` B 0 p ` 1 B 1 p ` 2 B 2 p ` 3 B 3 p B ` 1 B ` B ` +1 M ` +1 = n p ` +1 B 0 p ` B 1 p ` 1 B 2 p ` 2 B 3 p 2 B ` 1 p B ` B ` +1 ... M ` + k = n p ` + k B 0 p ` + k 1 B 1 p ` + k 2 B 2 p ` + k 3 B 3 p k +1 B ` 1 p k B ` B ` +1 ... 30

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Table4-2.Visualizingthe R -submodules N j ( ) SubmoduleBasis ( = N ` +1 =) N ` = n C 0 C 1 C 2 C 3 C ` 1 C ` N ` 1 = n C 0 C 1 C 2 C 3 C ` 1 p C ` N ` 2 = n C 0 C 1 C 2 C 3 p C ` 1 p 2 C ` ... N 3 = n C 0 C 1 C 2 C 3 p ` 4 C ` 1 p ` 3 C ` N 2 = n C 0 C 1 C 2 p C 3 p ` 3 C ` 1 p ` 2 C ` N 1 = n C 0 C 1 p C 2 p 2 C 3 p ` 2 C ` 1 p ` 1 C ` N 0 = n C 0 p C 1 p 2 C 2 p 3 C 3 p ` 1 C ` 1 p ` C ` whereonlythelastinclusionisnecessarilystrict.Choose abasis B ` +1 of ker( ) and extendittoabasis B ` [ B ` +1 of M ` ( ) .Continueinthisfashiontogetabasis [ ` +1 i =0 B i of M 0 ( ) .Nowlifttheelementsof B ` +1 toaset B ` +1 ofpreimagesin ker( ) .Continuing, ateachstageweenlarge B i +1 byadjoiningaset B i ofpreimagesin M i ( ) of B i .By Nakayama'sLemma,theset B = ` +1 [ i =0 B i isan R -basisof R m Noticethat N ` ( )= N ` +1 ( )= .Set N 0 = N ` ( ) .Then N 0 iscalledthe purication of Im ,andisthesmallest R -moduledirectsummandof R n containing Im .The elementarydivisorsof remainthesameifwechangethecodomainof to N 0 .Choose abasis C 0 of N 0 ( ) andextendittoabasis C 0 [ C 1 of N 1 ( ) .Continueinthisfashionto getabasis [ `i =0 C i of N ` ( ) .Nowwelifttheelementsof C 0 toaset C 0 ofpreimagesin N 0 ( ) .Continuing,ateachstageweenlarge C i byadjoiningaset C i +1 ofpreimagesin N i +1 ( ) of C i +1 .ByNakayama'sLemma,theset C 0 = ` [ i =0 C i isan R -basisof N 0 .Wethenset C = ` +1 [ i =0 C i 31

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tobeany R -basisof R n obtainedbyadjoiningto C 0 someset C ` +1 Lemma4.3. 1. Thebasis B constructedaboveisaleftSNFbasisfor 2. Thebasis C constructedaboveisarightSNFbasisfor Proof. For x 2B i 0 i ` ,considertheelement y = p i ( x ) 2 N 0 .Thecollectionof allsuchelementsformalinearlyindependentset,sincethe basis B extendsthebasis B ` +1 of ker( ) .Let Y denotethe R -submodulegeneratedbytheseelements.From Lemma 4.1 weseethattheindexof Im in Y isthesameastheindexof Im in N 0 Hence Y = N 0 ,andsotheseelementsformabasisof N 0 .Thematrixof withrespect to B andanybasisof R n obtainedbyextendingthisbasisof N 0 willthenbeindiagonal form.Thisprovespart( 1 ). Now,foreach y 2C i 0 i ` ,chooseanelement x 2 M i ( ) suchthat ( x )= p i y .Let X denotethe R -submoduleof R m generatedbytheseelements.The imagesoftheseelementsarecertainlylinearlyindependen t,hence X \ ker( )= f 0 g ByLemma 4.1 weseethat Im and ( X +ker( )) havethesameindexin N 0 .Therefore R m = X ker( ) ,andadjoininganybasisof ker( ) tothesegeneratorsof X givesa basisof R m .Withrespecttothisbasisand C ,thematrixof isindiagonalform. 32

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CHAPTER5 PROOFSOFTHEOREMS 5.1ProofofTheorem 3.1 Forbrevity,an r -dimensionalsubspaceof V willusuallyjustbecalledan r -subspace inwhatfollows.Sincealloftheelementarydivisorsof A arepowersof p ,wemightas wellview A asamatrixoverthe p -adicintegers Z p .Noneoftheelementarydivisor multiplicitiesareaffectedifwedothis,andwemayappealt oourresultsinChapter 4 A representsahomomorphismoffree Z p -modules Z L 2 p Z L 2 p thatsendsa 2 -subspacetothe(formal)sumofthe 2 -subspacesincidentwithit.We abusenotationbyusingthesamesymbolforboththematrixan dthemap.Wealso applyourmatricesandmapsontheright(so AB means“do A rst,then B ”). Let 1 = P x 2L 2 x andset Y 2 = n X x 2L 2 a x x 2 Z L 2 p X x 2L 2 a x =0 o Since jL 2 j isaunitin Z p ,wehavethedecomposition Z L 2 p = Z p 1 Y 2 WenowproveTheorem 3.1 .Themap A respectstheabovedecompositionof Z L 2 p andthuswegetalloftheelementarydivisorsof A bycomputingthoseoftherestriction of A toeachsummand.Since ( 1 ) A = q 4 1 ,weseethat e 4 t ( A )= e 4 t ( A j Y 2 )+1 and e i ( A )= e i ( A j Y 2 ) for i 6 =4 t Rewritingequation( 3–1 )weget A ( A +( q 2 q ) I )= q 3 I +( q 4 q 3 ) J 33

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andifwenowrestrict A to Y 2 ,theaboveequationreads A ( A +( q 2 q ) I )= q 3 I Let P and Q beunimodulartransformationssothat D = PAQ 1 actsdiagonallyon Y 2 Thenwegettherelation Q ( A +( q 2 q ) I ) P 1 = q 3 D 1 (5–1) whichgivestheSmithnormalformof A +( q 2 q ) I on Y 2 .Itfollowsfromthisequation that e i ( A j Y 2 )=0 for i > 3 t ,andso e 4 t ( A )=1 ,establishingpart( 1 )ofthetheorem(and mostofpart( 2 )). Italsofollowsimmediatelyfromequation( 5–1 )that e i ( A j Y 2 +( q 2 q ) I )= e 3 t i ( A j Y 2 ) (5–2) for 0 i 3 t .Since A j Y 2 A j Y 2 +( q 2 q ) I (mod p t ) ,wehavefromLemma 4.2 that e i ( A j Y 2 )= e i ( A j Y 2 +( q 2 q ) I )= e 3 t i ( A j Y 2 ) (5–3) for 0 i < t ,whichispart( 3 )ofthetheorem. Itremainstoproveparts( 4 )and( 5 )ofthetheorem,andalsothestatementfrom part( 2 )that e i ( A )=0 for t < i < 2 t .Denoteby V the -eigenspacefor A (asamatrix over Q p ,the p -adicnumbers).Then V q \ Z L 2 p and V q 2 \ Z L 2 p arepure Z p -submodulesof Y 2 .Noticethat V q \ Z L 2 p N t ( A j Y 2 ) and V q 2 \ Z L 2 p M 2 t ( A j Y 2 ) .Therefore, q 4 + q 2 =dim F p ( V q \ Z L 2 p ) dim F p N t ( A j Y 2 )= t X i =0 e i ( A j Y 2 ) and q 3 + q 2 + q =dim F p ( V q 2 \ Z L 2 p ) dim F p M 2 t ( A j Y 2 )= 3 t X i =2 t e i ( A j Y 2 ). 34

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Since ( q 4 + q 2 )+( q 3 + q 2 + q )=dim F p Y 2 ,theaboveinequalitiesareactually equalities andtheremainingelementarydivisormultiplicitiesmustb ezero.Thiscompletesthe proofofTheorem 3.1 Remark. Theaboveproofsimplyexploitsequation( 3–1 ),andmakesnouseofthe geometryof PG(3, q ) .ThereforeTheorem 3.1 isalsotruefortheadjacencymatrix A of anystronglyregulargraphwiththesameparameters. Theorem 3.2 willfollowfromamoregeneralresultwhichweprovebelow.H erewe explaintheconnectionbetweenthesetheorems.Let B denotetheincidencematrixwith rowsindexedby L 1 andcolumnsindexedby L 2 ,whereincidenceagainmeanszero intersection. B t denotesthetransposeof B ,andisjusttheincidencematrixoflinesvs. points.Itiseasytocheckthat B t B =( q 3 + q 2 ) I +( q 3 + q 2 q 1) A +( q 3 + q 2 q )( J A I ). (5–4) Justlikewith A ,wedenotealsoby B and B t theincidencemapsthesematrices representover Z p .Noticethat ( 1 ) B t B = q 4 ( q 2 + q +1)( q +1) 1 ,andsofor i 6 =4 t we have e i ( B t B )= e i ( B t B j Y 2 ) .Thusagainweconcentrateonthesummand Y 2 Wecanrewritetheequation( 5–4 )as B t B = [ A +( q 2 q ) I ]+ q 2 I +( q 3 + q 2 q ) J anduponrestrictionofmapsto Y 2 itreads B t B = [ A +( q 2 q ) I ]+ q 2 I ApplyingLemma 4.2 wehave,for 0 i < 2 t e i ( B t B j Y 2 )= e i ( A j Y 2 +( q 2 q ) I ). (5–5) Usingequation( 5–2 ),andconsideringonlynonzeromultiplicities,wethenget e 2 t + i ( A )= e t i ( B t B ), for 0 i t .(5–6) 35

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ThereforetoproveTheorem 3.2 itissufcienttocomputethe( p -adic)elementary divisorsofthematrix B t B .Thenaltheorembelowdescribesthese.Wecanactuallydo thisatthelevelofgeneralitymentionedintheintroductio n. 5.2TheGeneralResult Fortheremainderofthework, V isan ( n +1) -dimensionalvectorspaceover F q ,where q = p t isaprimepower. A r s isthe jL r jjL s j incidencematrixwithrows indexedbythe r -subspacesof V andcolumnsindexedbythe s -subspacesof V ,and twosubspacesareincidentifandonlyiftheirintersection istrivial.Wewillcomputethe elementarydivisorsof A r ,1 A 1, s asamatrixover Z p Let H denotethesetof t -tuplesofintegers ~ s =( s 0 ,..., s t 1 ) thatsatisfy,for 0 i t 1 1. 1 s i n 2. 0 ps i +1 s i ( p 1)( n +1) withsubscriptsreadmodulo t .Firstintroducedin[ 6 ],theset H waslaterusedin[ 2 ]to describethemodulestructureof F L 1 q undertheactionof GL( n +1, q ) .Fornonnegative integers ,denethesubsetsof H H ( s )= n ( s 0 ,..., s t 1 ) 2H t 1 X i =0 max f 0, s s i g = o and H ( r )= f ( n +1 s 0 ,..., n +1 s t 1 ) j ( s 0 ,..., s t 1 ) 2H ( r ) g = n ( s 0 ,..., s t 1 ) 2H t 1 X i =0 max f 0, s i ( n +1 r ) g = g o Toeachtuple ~ s 2H weassociateanumber d ( ~ s ) asfollows.For ~ s =( s 0 ,..., s t 1 ) 2 H denetheintegertuple ~ =( 0 ,..., t 1 ) by i = ps i +1 s i (subscriptsmod t ) 36

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Foraninteger k ,set d k tobethecoefcientof x k intheexpansionof (1+ x + + x p 1 ) n +1 Explicitly, d k = b k = p c X j =0 ( 1) j n +1 j n + k jp n Finally,set d ( ~ s )= Q t 1 i =0 d i Theorem5.1. Let e i ( A r ,1 A 1, s ) denotethemultiplicityof p i asa p -adicelementarydivisor of A r ,1 A 1, s 1. e t ( r + s ) ( A r ,1 A 1, s )=1 2. For i 6 = t ( r + s ) e i ( A r ,1 A 1, s )= X ~ s 2 ( i ) d ( ~ s ), where ( i )= [ + = i 0 t ( s 1) 0 t ( r 1) H ( r ) \H ( s ). Summationoveranemptysetisinterpretedtoresultin 0 Itwillbetechnicallyconvenienttoactuallyworkoveralar gerringthan Z p .Let K = Q p ( ) betheuniqueunramiedextensionofdegree t ( n +1) over Q p ,where is aprimitive ( q n +1 1) th rootofunityin K .Weset R = Z p [ ] tobetheringofintegers in K .Then R isadiscretevaluationring, p 2 R generatesthemaximalideal,and F = R = pR = F q n +1 .Fortheremainderofthisworkweviewallmatrixentriesasc oming from R Set G =GL( n +1, q ) .Uponxingabasisof V thereisanaturalactionof G onthe sets L i ,andinthisway R L i becomesan RG -permutationmodule.Asbefore, A r s will denoteboththematrixandtheincidencemap R L r R L s thatsendsan r -subspacetothe(formal)sumof s -subspacesincidentwithit.Sincethe actionof G preservesincidence,the A r s are RG -modulehomomorphisms.Clearlythe 37

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M i ( A r s ) and N j ( A r s ) are RG -modules.Wehavethe RG -decompositions R L k = R 1 Y k where 1 = P x 2L k x and Y k isthekernelofthesplittingmap X x 2L k a x x 7! 1 jL k j X x 2L k a x 1 andallthe A r s respectthesedecompositions.Reduction (mod p ) inducesahomomorphism of FG -permutationmodules F L r F L s whichwedenoteby A r s LetusindicatehowwewillproveTheorem 5.1 .Supposethatweareabletond unimodularmatrices P Q ,and E suchthat PA r ,1 E 1 = D r ,1 and EA 1, s Q 1 = D 1, s wherethematricesontherightarediagonal.Thenthesediag onalentriesarethe elementarydivisorsoftherespectivematrices A r ,1 and A 1, s .Sincethen PA r ,1 A 1, s Q 1 = D r ,1 D 1, s wewillthenhaveobtainedtheelementarydivisorsofthepro ductmatrix(providedthat wehavedetailedenoughknowledgeoftheelementarydivisor softhefactormatrices). Example5.1. Considerthematrixproduct 0B@ p 1 0 p 1CA 0B@ 1 p 0 p 2 1CA = 0B@ p 0 0 p 3 1CA 38

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ThisproductmatrixisalreadyinSmithnormalform.Notetha tbothofthefactormatrices areequivalentto 100 p 2 .Evenallowingpermutationofthediagonalentries,itisno t possiblefortwosuchdiagonalmatricestomultiplyto p 0 0 p 3 Ingeneralitisnotpossibletondsuchamatrix E ,astheaboveexampleshows. Thuswhentryingtondtheelementarydivisorsofamatrixpr oduct,knowledgeof theelementarydivisorsofthefactormatricesisingeneral notsufcient(formore informationonthisinterestingtopic,see[ 11 ]and[ 14 ]).Thereforeweshouldnotexpect suchanintermediatematrix E toexistinoursituation.Yetitdoes!Toseewhatisso specialhere,passfrommatricesbacktomodules.Thekeying redientisthestructure of R L 1 asan RG -module,asthematrix E arisesfromchoosingabasisof R L 1 thatis “compatible”withbothfactormaps.Wealreadyhavethecorr ectterminologyforthis. Lemma5.1. Thereexistsabasis B of R L 1 thatissimultaneouslyaleftSNFbasisfor A 1, s andarightSNFbasisfor A r ,1 Proof. Thegroup G hasacyclicsubgroup S whichisisomorphicto F .Since R containsaprimitive j S j th rootofunity,itfollowsthat K isasplittingeldfor S andthat theirreducible K -charactersof S taketheirvaluesin R .Let S denotethequotientof S bythesubgroupofscalarmatrices.Then S actsregularlyon L 1 ,and j S j = jL 1 j isaunit in R .Therefore,foreachcharacter of S ,thegroupring R S containsanidempotent element h thatprojectsontothe(rankone) -isotypiccomponentof R L 1 .Wethus obtainan R -basis B = f v j 2 Hom( S R ) g of R L 1 ,where v 2 h R L 1 suchthat p 6j v NowletusconstructaleftSNFbasisfor A 1, s ,inthemannerandnotationdescribed inSection 4.2 .Sinceeach F S -submoduleof F L 1 isadirectsumoftheisotypic componentsthatitcontains,weseethatwecantakeeachofth esets B i tobeasubset of B .Supposewelift v 2 B i toanelement f 2 M i ( A 1, s ) .Writing f = X 2 Hom( S R ) c v 39

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weseethat f = c v andso c mustbeaunitin R .Since M i ( A 1, s ) isan R S -submodule, wehavethat h f = c v isalsoin M i ( A 1, s ) .Thisprovesthatwemaychoosetolift v to v intheconstruction,andthat B isaleftSNFbasisfor A 1, s Anidenticalargument(liftingeach v intosome N j ( A r ,1 ) )showsthat B isaright SNFbasisfor A r ,1 Itremainstoshowthattheelementarydivisormultipliciti esareasstatedinthe theorem.Firstweneedamoreprecisedescriptionofthe FG -submodulelatticeof F L 1 .Thefactsthatweneedareasfollows(see[ 2 ,TheoremA]). F L 1 = F 1 Y 1 isa multiplicity–free FG -module,andthe FG -compositionfactorsof Y 1 areinbijectionwith theset H .Thedimensionover F ofthecompositionfactorcorrespondingtothetuple ~ s is d ( ~ s ) .Moreover,ifwegive H thepartialorder ( s 0 ,..., s t 1 ) ( s 0 0 ,..., s 0 t 1 ) () s i s 0 i foralli thenthe FG -submodulelatticeof Y 1 isisomorphictothelatticeoforderidealsof H andthetuplescontainedinanorderidealcorrespondtothec ompositionfactorsofthe respectivesubmodule.Thusitisclearwhatismeantbythest atementthatasubquotient of Y 1 determines asubsetof H Remarks. 1. Theeld k in[ 2 ]isactuallyanalgebraicclosureof F q ,but(asobservedin[ 4 ]) itfollowsfrom[ 2 ,TheoremA]thatall kG -submodulesof k L 1 aresimplyscalar extensionsof F q G -modules,andtherefore[ 2 ,TheoremA]isalsotrueoveroureld F = F q n +1 .Thisobservationalsopermitsustomakeuseofcertainresu ltsfrom[ 4 ], wheretheeldis F q 2. Thedetailedinformationthatweneedabouttheelementaryd ivisorsof A r ,1 and A 1, s weobtainfrom[ 4 ].Itshouldalsobenotedthattheincidencerelation consideredin[ 2 4 13 ]is nonzero intersection(i.e.,thecomplementaryrelation wheretwosubspacesareincidentifandonlyiftheirinterse ctionisnontrivial).If A 0r s isthecorrespondingincidencematrixfornonzerointersec tion,thenwehave A r s = J A 0r s 40

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Inparticular, A r s j Y r = A 0r s j Y r Thereforethe( p -adic)Smithnormalformsof A r s and A 0r s candifferonlywith respecttowheretheymap 1 .Thisaccountsfortheextratermappearinginthe calculationof p -ranksin[ 2 4 13 ]. Lemma5.2. 1. The FG -module M ( A 1, s j Y 1 ) = M +1 ( A 1, s j Y 1 ) determinesthesubset H ( s ) 2. The FG -module N ( A r ,1 j Y r ) = N 1 ( A r ,1 j Y r ) determinesthesubset H ( r ) Proof. Part( 1 )isthecontentof[ 4 ,Theorem3.3](seeRemarksabove).Inorderto prove( 2 ),rstobservethatforeach k L k isanorthonormalbasisforanondegenerate G -invariantsymmetricbilinearform h i k on R L k .Usetheinducedformon F L k to identifyeachpermutationmodulewithitsdual(contragred ient)module,andobservethat A s r isthedualmapinducedby A r s .Sincethetuples ( s 0 ,..., s t 1 ) and ( n +1 s 0 ,..., n + 1 s t 1 ) aredeterminedbydualcompositionfactors[ 2 ,Lemma2.5(c)],part( 2 )will followimmediatelyifwecanshowthe FG -moduleisomorphism N ( A r s j Y r ) = N 1 ( A r s j Y r ) = M ( A s r j Y s ) = M +1 ( A s r j Y s ). Itissufcienttoshowthat N ( A r s j Y r ) ? = M +1 ( A s r j Y s ). Weproceedbyinductionon .When =0 ,wehave N 0 ( A r s j Y r ) ? = f y j y 2 Y s h ( x ) A r s y i s 0(mod p ) forall x 2 Y r g = f y j y 2 Y s h x ,( y ) A s r i r 0(mod p ) forall x 2 Y r g = M 1 ( A s r j Y s ). 41

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wherethelastequalityfollowsfromthenondegeneracyofth einducedformon Y r .Now assume > 0 .Itiseasytocheckthat M +1 ( A s r j Y s ) N ( A r s j Y r ) ? .Wethenhave M +1 ( A s r j Y s ) N ( A r s j Y r ) ? N 1 ( A r s j Y r ) ? = M ( A s r j Y s ), withtheequalitybyourinductionhypothesis.Sinceclearl y e ( A s r j Y s )= e ( A r s j Y r ) ,it nowfollowsfromLemma 4.1 andtheaboveinclusionsthat M +1 ( A s r j Y s )= N ( A r s j Y r ) ? 2 H (2) H 0 (2) 1 H (2) 33 32 y y 23 E E 31 y y 22 y y E E 13 E E 21 y y E E 12 y y E E 11 y y E E 33 32 y y 23 E E 31 y y 22 y y E E 13 E E 21 y y E E 12 y y E E 11 y y E E H 1 (2) 0 H (2) H 2 (2) Figure5-1.IllustratingLemma 5.2 when n =3 and r = s = t =2 ProofofTheorem 5.1 Fixan FG -compositionseries f 0 g F 1 = U 0 U 1 F L 1 Startingwiththe F -basis f v 1 S g of U 0 ,wecanextendthisusingelementsof B toabasis of U 1 .Continuinginthisfashion,wethusgetthedisjointunion B = f v 1 S g[D 1 [ where D i aretheelementsof B extending U i 1 to U i .Itisclearthateachquotient U i = U i 1 ( i 1 )isisomorphicasan F S -moduletothe F S -submoduleof Y 1 spannedby D i .Ifthesimple FG -module U i = U i 1 determinesthetuple ~ s 2H ,thenwesaywillsay thateachelementof D i determines thetuple ~ s .Thisassignmentofelementsof B to tuplesin H iswell–denedindependentoftheabovecompositionseries ,asfollowsfrom 42

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thefactthattheisomorphismtypeofan F S -submoduleof Y 1 iscompletelydetermined bythecharactersitaffords. ByLemma 5.2 ,thetupledeterminedby v belongsto H ( s ) \ H ( r ) preciselywhen thefollowingtwoconditionshold: 1. v 2 M ( A 1, s j Y 1 ) but v = 2 M +1 ( A 1, s j Y 1 ) 2. v 2 N ( A r ,1 j Y r ) but v = 2 N 1 ( A r ,1 j Y r ) Itimmediatelyfollowsthat e i ( A r ,1 A 1, s j Y r )= X + = i X ~ s 2H ( s ) \ H ( r ) d ( ~ s ), for i 0. Since H ( s )= ; for > t ( s 1) and H ( r )= ; for > t ( r 1) ,wehave e i ( A r ,1 A 1, s j Y r )=0, for i > t ( r + s 2). Wewillusethe q -binomialcoefcients 264 m ` 375 q = ( q m 1)( q m 1 1) ( q m ` +1 1) ( q 1)( q 2 1) ( q ` 1) fornon-negativeintegers m and ` with m ` .Then ( 1 ) A r ,1 A 1, s = q r + s [ n r ] q n +1 s 1 q 1 andwehave e t ( r + s ) ( A r ,1 A 1, s )=1 ,completingtheproofofTheorem 5.1 Remark. Since d ( ~ s )=0 for ~ s 2 [ n ] t nH ,thereisnoeffectonthenumericalresultof Theorem 5.1 ifwereplace H with [ n ] t inthenotationprecedingthestatementofthe theorem. ProofofTheorem 3.2 Werecoverouroriginalsituationbysetting n =3 and r = s =2 (so A 2,2 = A and A 1,2 = B ).Replace H with [3] t inthenotationprecedingTheorem 5.1 43

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Thenitiseasytoseethat H (2)= f ~ s 2 [3] t j ~ s containsexactly ones g and H (2)= f ~ s 2 [3] t j ~ s containsexactly threes g Hence ( i )= [ + = i H (2) \ H (2) = f ~ s 2 [3] t j ~ s containsexactly t i twos g = H ( t i ). Therefore,for 0 i t e t i ( B t B )= X ~ s 2H ( i ) d ( ~ s ) andinviewofequation( 5–6 )weseethatTheorem 3.2 followsfromTheorem 5.1 Asmentionedintheintroduction,theproblemofcomputingt heintegerinvariants of A r s ingeneralisstillverymuchunsolved.The p -ranksoftheincidencematrices A r s werecomputedin[ 13 ],andobservethatthe p -rankofanintegermatrixisjustthe multiplicityof p 0 asa p -adicelementarydivisor.Weconcludewiththefollowingea sy corollaryofTheorem 5.1 Corollary5.2. NotationisthatofTheorem 5.1 .Let e i ( A r s ) denotethemultiplicityof p i asa p -adicelementarydivisorof A r s .Then,for 0 i < t e i ( A r s )= X ~ s 2 ( i ) d ( ~ s ). Proof. Let x 2L r .Then ( x ) A r s = X y 2L s a x y y 44

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where a x y = jf z 2L 1 j z \ x = f 0 g and z \ y = f 0 ggj = 8>><>>: [ n +1 1 ] q [ r 1 ] q [ s 1 ] q if x \ y 6 = f 0 g [ n +1 1 ] q [ r 1 ] q [ s 1 ] q + [ k 1 ] q if dim( x \ y )= k 1. Then a x y 1(mod q ) when x \ y = f 0 g and q divides a x y otherwise.Hence A r ,1 A 1, s A r s (mod p t ) andthecorollarynowfollowsfromLemma 4.2 45

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APPENDIX:SAMPLESAGEPROGRAM x=walltime()p=2t=2q=p^tF.=GF(q)V=VectorSpace(F,4)S=tuple(V.subspaces(2))l=len(S)print"nowformingincidencematrixA"A=matrix(ZZ,l)foriinrange(l-1): forjinrange(i+1,l): ifdim(S[i].intersection(S[j]))==0: A[i,j]=1 A=A+A.transpose()y=walltime(x)print"took",y,"seconds"save(A,'./programs/Results-4dim/incmat2^2') 46

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REFERENCES [1] E.F.Assmus,Jr.andJ.D.Key. Designsandtheircodes ,volume103of Cambridge TractsinMathematics .CambridgeUniversityPress,Cambridge,1992. [2] MatthewBardoeandPeterSin.Thepermutationmodulesfor GL( n +1, F q ) acting on P n ( F q ) and F n 1 q J.LondonMath.Soc.(2) ,61(1):58–80,2000. [3] A.E.Brouwer,A.M.Cohen,andA.Neumaier. Distance-regulargraphs ,volume18 of ErgebnissederMathematikundihrerGrenzgebiete .Springer-Verlag,Berlin, 1989. [4] DavidB.Chandler,PeterSin,andQingXiang.Theinvariantf actorsoftheincidence matricesofpointsandsubspacesin PG( n q ) and AG( n q ) Trans.Amer.Math. Soc. ,358(11):4935–4957(electronic),2006. [5] J.W.L.Glaisher.HenryJohnStephenSmith. MNRAS ,44(6):138–149,1884. [6] NoboruHamada.Onthe p -rankoftheincidencematrixofabalancedorpartially balancedincompleteblockdesignanditsapplicationstoer rorcorrectingcodes. HiroshimaMath.J. ,3:153–226,1973. [7] J.W.P.Hirschfeld. Finiteprojectivespacesofthreedimensions .Oxford MathematicalMonographs.TheClarendonPressOxfordUnive rsityPress,New York,1985.OxfordSciencePublications. [8] ThomasW.Hungerford. Algebra ,volume73of GraduateTextsinMathematics Springer-Verlag,NewYork,1980.Reprintofthe1974origin al. [9] EricS.Lander. SymmetricDesigns:AnAlgebraicApproach .CambridgeUniversity Press,1983.LondonMath.Soc.LectureNotes74. [10] MorrisNewman. Integralmatrices .AcademicPress,NewYork,1972.Pureand AppliedMathematics,Vol.45. [11] JosephJohnRushanan. Topicsinintegralmatricesandabeliangroupcodes ProQuestLLC,AnnArbor,MI,1986.Thesis(Ph.D.)–Californ iaInstituteof Technology. [12] PeterSin.Theelementarydivisorsoftheincidencematrice sofpointsandlinear subspacesin P n ( F p ) J.Algebra ,232(1):76–85,2000. [13] PeterSin.The p -rankoftheincidencematrixofintersectinglinearsubspa ces. Des. CodesCryptogr. ,31(3):213–220,2004. [14] KyleD.Wallace.Extensionofmappingsinniteabeliangrou ps. Amer.Math. Monthly ,79:622–624,1972. [15] RichardM.Wilson.Adiagonalformfortheincidencematrice sof t -subsetsvs. k -subsets. EuropeanJ.Combin. ,11(6):609–615,1990. 47

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[16] QingXiang.Recentprogressinalgebraicdesigntheory. FiniteFieldsAppl. 11(3):622–653,2005. [17] QingXiang.Recentresultson p -ranksandSmithnormalformsofsome 2 ( v k ) designs.In Codingtheoryandquantumcomputing ,volume381of Contemp.Math. pages53–67.Amer.Math.Soc.,Providence,RI,2005. 48

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BIOGRAPHICALSKETCH JoshuaEvansDuceywasbornin1983inHonolulu,Hawaii.Hegr ewupinHanover, Maryland,justafewmilesfromtheNationalSecurityAgency .Aftergraduatingfrom MeadeSeniorHighSchoolin2001,JoshattendedcollegeinVi rginiaattheUniversity ofRichmond.There,hestudiedmathematicsandphilosophy, earninghisbachelor's degreeintheformerin2005.Itwasinthemiddleofhissenior yearatRichmondthathe decidedthathewantedtobeamathematician.Afterbuilding boatdocksforasummer inSt.Petersburg,Florida,Joshenrolledingraduateschoo lattheUniversityofFlorida. Itwasherethathemethisanc ee,MinahOh.In2011,Joshearnedhisdoctoratein mathematicsundertheguidanceofDr.PeterSin. 49