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Some Algebraic Problems From Coding Theory

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Title:
Some Algebraic Problems From Coding Theory
Creator:
Arslan, Ogul
Place of Publication:
[Gainesville, Fla.]
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University of Florida
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Language:
english
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1 online resource (46 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Sin, Peter
Committee Members:
Drake, David A.
Tiep, Pham H.
Turull, Alexandre
Keating, Kevin
Sitharam, Meera
Graduation Date:
8/8/2009

Subjects

Subjects / Keywords:
Binary codes ( jstor )
Eigenfunctions ( jstor )
Geometric lines ( jstor )
Geometry ( jstor )
Integers ( jstor )
Mathematical vectors ( jstor )
Mathematics ( jstor )
Matrices ( jstor )
Polynomials ( jstor )
Vector spaces ( jstor )
Mathematics -- Dissertations, Academic -- UF
coding, finite, generalized, geometry, incidence, ldpc, lumq, matrix, prank, quadrangle, representation, symplectic, theory
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Electronic Thesis or Dissertation
born-digital ( sobekcm )
Mathematics thesis, Ph.D.

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Abstract:
SOME ALGEBRAIC PROBLEMS FROM CODING THEORY Let F be a finite field of size q and characteristic p. A low density parity check (LDPC) code is a finite dimensional subspace of a vector space over F. A parity check matrix of an LDPC code is a binary sparse matrix which is orthogonal to the code. In this work, we describe a family of LDPC codes called the LU(3,q) codes over F . Let M (P,L) be the point-line incidence matrix of the symplectic generalized quadrangle. We give a description of a submatrix H of M(P,L) such that, any LU(3,q) code has either H or the transpose of H as its parity check matrix. Previously, Peter Sin and Qing Xiang derived a formula for the dimension of the LU(3,q) codes for the case where F has an odd characteristic. If F has an even characteristic , the field of the geometry and the parity check matrix have the same characteristic, hence the solution requires different techniques. In this research, we give a descriptions of the points and lines of the symplectic generalized quadrangle using characteristic functions and polynomials. Using representation theory of the symplectic group SP(4,q), we find a basis for the column space of M(P,L). We use this result to show that the 2-rank of H is rank_2(M(P,L))-2q. Hence, the dimension of an LU(3,2^t) code is q^3+2q-rank_2(M(P,L)). This completes the dimension problem for the LU(3,q) codes. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2009.
Local:
Adviser: Sin, Peter.
Statement of Responsibility:
by Ogul Arslan.

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University of Florida
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University of Florida
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Copyright Arslan, Ogul. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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Classification:
LD1780 2009 ( lcc )

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Foremost,IwouldliketoexpressmygratitudetomyadvisorProfessorPeterSin,forhisguidanceandinspiration.Hewasalwaysthere,withhisimmenseknowledgeandenthusiasm,wheneverIneededhelp.Thisresearchwouldnotbepossiblewithouthiscontinuoussupport,patience,motivationandencouragement.Iamgratefultotherestofmysupervisorycommittee:Dr.DavidDrake,Dr.KevinKeating,Dr.MeeraSitharam,Dr.PhamHuTiepandDr.AlexandreTurull,fortheirmentoringandencouragement.IgivemysincerethankstoallmyprofessorsinbothWayneStateUniversityandtheUniversityofFlorida,forinspiringme.IalsogivemythankstocreatorsofSAGE(MathematicalSoftwareVersion3.4.1, 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 6 ABSTRACT ........................................ 7 CHAPTER 1FINITEGEOMETRIES ............................... 8 1.1IncidenceStructures .............................. 8 1.2ProjectiveSpaces ................................ 9 1.3FiniteGeneralizedQuadrangles ........................ 11 2LOWDENSITYPARITYCHECK(LDPC)CODES ............... 18 2.1LinearCodes .................................. 18 2.2Low-DensityParityCheckCodes ....................... 19 3LU(m;q)CODES ................................... 21 3.1IncidenceStructure(q) ............................ 21 3.2LU(m,q)Codes ................................. 22 3.2.1LU(2,q)Codes .............................. 23 3.2.2LU(3,q)Codes .............................. 24 3.2.3LU(m,q)Codesform>3 ........................ 25 4DIMENSIONSOFLU(3;q)CODES ........................ 26 4.1AnotherDescriptionforLU(3,q)Codes .................... 26 4.2DimensionsforC(P,L)andaLowerBoundfortheDimensionofLU(3,q) 27 4.3GridsofLines .................................. 31 4.4ApproachbyUsingPolynomials ........................ 36 4.5DigitizablePolynomialsinR 39 4.6OntheKerneloftheProjectionMap ..................... 40 REFERENCES ....................................... 44 BIOGRAPHICALSKETCH ................................ 46 5

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Figure page 4-1LinesthatareincidentwithXandY. ....................... 30 4-2Lines0and0. .................................... 32 4-3Morelinesinthegrid. ................................ 33 4-4Linesofthegrid ................................... 33 4-5Intersectionofand. ............................... 34 4-6Summinglinesinthegrid. .............................. 34 4-7Gridlinesbetween`and`0. ............................. 35 6

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LetFbeaniteeldofsizeqandcharacteristicp.Alowdensityparitycheck(LDPC)codeisanitedimensionalsubspaceofavectorspaceoverF.AparitycheckmatrixofanLDPCcodeisabinarysparsematrixwhichisorthogonaltothecode.Inthiswork,wedescribeafamilyofLDPCcodescalledtheLU(3,q)codesoverF.LetM(P,L)bethepoint-lineincidencematrixofthesymplecticgeneralizedquadrangle.WegiveadescriptionofasubmatrixHofM(P,L)suchthat,anyLU(3,q)codehaseitherHorthetransposeofHasitsparitycheckmatrix. Previously,PeterSinandQingXiangderivedaformulaforthedimensionoftheLU(3,q)codesforthecasewhereFhasanoddcharacteristic.IfFhasanevencharacteristic,theeldofthegeometryandtheparitycheckmatrixhavethesamecharacteristic,hencethesolutionrequiresdierenttechniques.Inthisresearch,wegiveadescriptionsofthepointsandlinesofthesymplecticgeneralizedquadrangleusingcharacteristicfunctionsandpolynomials.UsingrepresentationtheoryofthesymplecticgroupSP(4,q),wendabasisforthecolumnspaceofM(P,L).Weusethisresulttoshowthatthe2-rankofHisrank2(M(P;L))2q.Hence,thedimensionofanLU(3,2t)codeisq3+2qrank2(M(P;L)).ThiscompletesthedimensionproblemfortheLU(3,q)codes. 7

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12 ]formoredetailedinformationaboutprojectivespaces. NotethattheelementsofPG(n;K)aretheone-dimensionalsubspacesofVwiththeorigindeleted.ThesearecalledthepointsofPG(n;K). Forconvenience,ifweviewaone-dimensionalsubspaceofVasapointinPG(n;K)weautomaticallyassumethattheoriginisomitted.Conversely,ifweviewapointofPG(n;K)asaone-dimensionalsubspaceofV,thenweautomaticallyassumethattheoriginisaddedtotheelementsofthepoint. Arepresentativeofapointp2PG(n;K)isanonzerovectorvinpsuchthatp=hvi.Anothernotationisthehomogeneouscoordinates.Ifv=(v0;v1;:::;vn)isarepresentativeofapointp,wewritep=[v0:v1:::::vn].Twopointsarecalledlinearlyindependentiftheirrepresentativevectorsarelinearlyindependent. 9

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Itisnothardtoseethat,ifK=Fqforsomeprimepowerqand0rn,thenthenumberofsubspacesofVofdimensionris (qn+11)(qn1):::(qnr+21) (qr1)(qr11):::(q1):(1{1) Inparticular,thenumberofonedimensionalsubspacesofVis LetPandLdenotethesetsofpointsandlinesofPG(n;K).Wesaythatapointp2Pisincidentwithaline`2L,ifthesubspaceofpiscontainedinthesubspaceof`.Inthiscase,wewritep2`.Thus,thepair(P;L)withtherelationofinclusionisanincidencesystem.Furthermore,if0rnwecantalkabouttheincidencesystems(P;Lr),whereLristhesetofr-spacesofPG(n;q). Oneofthenaturalquestionstoaskisthep-rankoftheincidencematricesofthesesystems.Thefollowingtheoremcanbefoundin[ 6 ].ItwasdeducedfromHamada'sarticlesin[ 10 ]and[ 11 ]. pcXi=0(1)in+1in+psj+1sjipn(1{3) 10

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ThisresultwasalsoprovenbyGrahamandMacWilliams[ 9 ]fortheplane,andbyGoethalsandDelsarte[ 8 ],MacWilliamsandMann[ 17 ],andbySmith[ 23 ]forgeneraln.Moregeneralversionoftheorem 1 isobtainedbyP.Sinin[ 21 ].Theauthorprovedaformulaforthep-rankoftheincidencematrixbetweenthed-dimensionalande-dimensionalsubspacesofVsuchthattheincidencerelationisthenon-trivialintersection. GivenanincidencestructureS=(P;B;I)theincidencegraphofSisdenedasfollows.ThevertexsetoftheincidencegraphisP[B.TheverticesinParepairwisedisconnected,similarlytheverticesinBarepairwisedisconnected.Twoverticesp2Pand`2BareconnectedbyanedgeifandonlyifpI`.Itiseasytoseethattheincidencegraphisabipartitegraph. 11

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25 ],alsoseeDembowski[ 5 ]. 2 ].Thedenitionofaquadranglecanberephrasedasfollows. 19 ]. ItiseasytoseethatifS=(P;B)isageneralizedquadrangleoforder(s;t),thenS=(B;P),thedualofS,isageneralizedquadrangleoforder(t;s). 12

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Letp?bethesetofpointsinallthelinesthroughpincludingpitself.Thatis, Similarly,wedene`?tobethesetofalllinesthatarecollinearto`.Thatis, Thetraceofapairofpoints(ortwolines)fp;p0gistheintersectionofp?andp0?.Wewritethisas Notethatifpp0andp6=p0,thentheonlypointsincommonbetweenp?andp0?aretheonesonthelinethroughpandp0.Hence,jfp;p0g?j=s+1.Ontheotherhand,ifp6p0thenbyquadrangleproperties,foreveryline`throughpthereisauniquelinepassingthroughp0andintersecting`.Sincetherearet+1linesthroughp,weconcludethatjp?\p0?j=t+1.Similarly,for`;`02B,if``0,thenjf`;`0g?j=t+1andotherwisejf`;`0g?j=s+1.Ingeneral,ifPP(respectively,ifLB),then 13

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Letp6=p0betwopoints.Then,thespanofthesepointsistheset Span(p;p0)=fp;p0g??=fp2Pjp2r?;8r2fp;p0g?g(1{10) Wesaythatapairofpoints(p;p0)isregularifeitherpp0andp6=p0orp6p0andjSpan(p;p0)j=t=1.Thepointpiscalledregularif(p;p0)isregularforallp02P,p6=p0. AlloftheclassicalgeneralizedquadranglescanbeembeddedintoPG(n;F).Moreovertheirautomorphismgroupscontainclassicalgroups.Therearethreemainclassesdependingonthegroupactingonthem. O(n+1;F)=fM2GL(n+1;q);F)jMTM=MMT=Ig(1{11) Wendthepointsandthelinesofthisquadrangleasfollows.DonateVwiththequadraticformQ:V!Fsuchthat,Q(v)=v0v1+v2v3forv=(v0;v1;v2;v3)2V;forn=3Q(v)=v20+v1v2+v3v4forv=(v0;v1;v2;v3;v4)2V;forn=4Q(v)=f(v0;v1)+v2v3+v4v5forv=(v0;v1;v2;v3;v4;v5)2V;forn=5 14

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Notethatwhenn=3,thequadrangleistrivial.Itisagrid. WedescribetheHermitianquadranglesasfollows.Thesetofpointsofthequadrangleis, AsubspaceofVissaidtobeisotropicifH(u;v)=0wheneveruandvarebothinthesubspace.Thelinesofthequadranglearethetotallyisotropic2-dimensionalsubspacesofV.Thatis, Theautomorphismgroupofthesequadranglescontainstheunitarygroups, U(n+1;q)=fU2GL(n+1;q)jUU=UU=Ig(1{15) 15

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TheHermitianquadrangleshavethefollowingproperties.Ifn=3thens=q2;t=q;jPj=(q2+1)(q3+1);jLj=(q+1)(q3+1).Ifn=4thens=q2;t=q3;jPj=(q2+1)(q5+1);jLj=(q3+1)(q5+1). ThesingularpointsandthetotallyisotropiclinesofPG(3;q)formstheSymplecticquadrangleW(q).TheautomorphismgroupofthisquadranglecontainstheSymplecticgroup, Sp(4;q)=fM2GL(4;q)jMTJM=Jg(1{17) whereJisanon-singular,skew-symmetricmatrix. 21 ]isstillunknown.Bychangingtheincidencesystemfromnonzerointersectiontosomethingelseonecanndmoreopenproblems. 16

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Denition7. TherankofthedualcodeC?isndim(C)=nk.AgeneratormatrixHofthedualcodeC?iscalledaparitycheckmatrixofthecodeC.Ifh:Fnq!FnkqisthelineartransformationoftheparitycheckmatrixH,thekernelofthistransformationisthecodeC.Wecandenetheparitycheckmatrixmoreformallyasfollows. Theweightofthecodewordxisthenumberofnon-zerocomponentsofx.Itisdenotedbyw(x).Henceitiseasytoseethatd(x;y)=w(yx). 18

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ItispossibletoprovethattheminimumdistanceistheleastnumberofcolumnsofH,theparitycheckmatrixofC,thatsumsupto0. Theminimumdistanceofacodegivesusaninformationabouthowmanyerrorscanbecorrectedaftertransmittingtheinformation.Therelationshipbetweenthenumberoferrorsthatcanbecorrectedandtheminimumdistanceisgiveninthefollowingtheorem. 2cerrors. 6 ],[ 3 ]or[ 18 ]. Therateofan(n;k)codeisthenumberofbitsperchannelusebeingtransmitted.Itistherationk=n.Inordertohavea'good'codeweexpectk=nanddtobelarge.Sogivenalinearcode,wewouldliketoknowisitsdimensionandminimumdistance.Ifthecodeisobtainedsystematicallythenitissomewhateasiertondthese.So,itisnaturaltolookatthecodesthatarisefromnitegeometries. 7 ]in1962.Theirdecodingperformanceisverygoodsotheybecamepopularrecently. 19

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ThecodeswewillstudyinthenextchapterareLDPCcodesobtainedfromthesymplecticgeneralizedquadrangleW(q). 20

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InthischapterwedescribetheLU(m;q)codes.Then,wetalkaboutthepropertiesofLU(2;q)andLU(3;q)codesandwegiveashortdiscussionaboutthepropertiesofLU(m;q)codesform>3. Letq=ptwherepisaprime.Weconstructthesemiplane(q)asfollows.LetPandLbetwoinnitedimensionalvectorspacesoverFq.Thus,anyelementinthesespacesisavectorwithinnitelymanycomponents.ThevectorspacesPandLarecalledthepointsandlinesofthesemiplane(q).Anypointp2Pwillbedenotedby(p),andanyline`2Lwillbedenotedby[`].Thatis,wewilluseparenthesesandbracketsinordertodistinguishbetweentheelementsofPandL.Wecanusethefollowingindexingforthepointsandlines.(p)=(p1;p1:1;p1:2;p2:1;p2:2;p02:2;p2:3;p3:2;p3:3;p03:3;:::;p0i:i;pi:i+1;pi+1:i;pi+1:i+1;:::) [`]=[`1;`1:1;`1:2;`2:1;`2:2;`02:2;`2:3;`3:2;`3:3;`03:3;:::;`0i:i;`i:i+1;`i+1:i;`i+1:i+1;:::](3{1) suchthat Apoint(p)isincidentwithaline[`]ifandonlyifthefollowingconditionsaresatisedfori=1;2;3;::: `i:ipi:i=`1pi1:i`0i:ip0i:i=p1`i:i1`i:i+1pi:i+1=p1`i:i`i+1:ipi+1:i=`1p0i:i 21

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Letm2beanintegerconsidertheincidencesystem(m;q)=(P(m);L(m);Im)whereP(m)andL(m)aresetofvectorsobtainedfromprojectingtheelementsofPandLontotherstmcomponents.For(p)2P(m),and[`]2L(m)wesay(p)isincidentwith[`]ifandonlyifrstmequationsincondition( 3{3 )aresatised.WeletD(m;q)denotetheincidencegraphof(m;q). Thefollowingpropositioncanbefoundin[ 15 ]. 15 ], (ii)Ifm2isanevenintegerthenthegirthofD(m;q)isatleastm+4[ 16 ]. 24 ]aretheD(m;q).Itisimmediatethatthesecodeshavelengthqm.Theotherpropertiesliketheminimumdistanceanddimensionvarybythechoiceofmandq.Thesecodeswererstintroducedin[ 13 ].TheauthorsalsoinvestigatedthepropertiesofLU(2;q)andLU(3;q)in[ 13 ].WhileLU(2;q)wascompletelydescribed,thedimensionsofLU(3;q)wereconjecturedforthecasewhereqisanoddprimepowerandtheothercaseremainedunknownforawhile.In2006,P.SinandQ.XiangprovedtheconjectureandgaveaformulaforthedimensionsofLU(3;q)codesforoddqin[ 22 ].Theyalsoobtainedalowerboundforthedimensionwhenqiseven.ThegivenboundistheactualdimensionbythecomputercalculationsofJ.-L.Kimuptoq=16.Weprovethelowerboundistheactual 22

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3{1 )weget and Theincidencerelationisthesatisfactionofthersttwoconditionsof( 3{3 ).Henceapoint(p1;p1:1)isincidentwithaline[`1;`1:1]ifandonlyif and aresatised.Notethat,withtheequationsin( 3{2 )thesetwoconditionsarethesame.Hence,wecandescribetheLU(2;q)codesasfollows. Suppose(P;L;I)isanincidencestructurewhere .Apoint(a;b)isincidentwithaline[x;y]ifandonlyify=ax+b.TheincidencematrixofthisstructureisdenotedbyH.ThebinaryLU(2;q)codesarethecodesobtainedbytheparitycheckmatricesHandHT. 13 ]. 23

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14 ]. 3{1 )weget and Wesaythatapoint(p1;p1:1;p1:2)isincidentwithaline[`1;`1:1;`1:2]ifandonlyiftherstthreeoftheconditions( 3{3 )aresatised.Thatis, Onceagainbecauseoftheequationsin( 3{2 )thersttwooftheseequationsareequivalent.HencewecandescribetheLU(3;q)codesasfollows. Let(P;L;I)beanincidencestructuresuchthat 24

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3{11 ),( 3{12 ),and( 3{13 )weobtainthatapoint(a;b;c)2Pisincidentwithaline[x;y;z]2Lifandonlyif LetHbetheincidencematrixofthisstructure.ThebinaryLU(3;q)codesarethelinearcodesobtainedfromtheparitycheckmatricesHandHT. Thethemoredetailedexplanationsaboutthefollowingpropertiescanbefoundin[ 13 ]. 3{3 )givesamorecomplicatedstructurefortheincidencerelation.However,sincethesegraphshavegirthsbiggerthan2dm=2e+4,wehavethefollowingtheoremaboutthelowerboundontheminimumdistance.

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InthischapterwegiveformulasforthedimensionsofLU(3;q)codes.Thisproblemissolvedintwocases,qoddandqeven. (v;v0)=x0y3x3y0+x1y2x2y1:(4{1) ConsidertheprojectivespaceP(V),thespaceofonedimensionalsubspacesofV.LetPbethesetofpointsofP(V),thatisthesetofonedimensionalsubspacesofV.WesometimesdenotetheelementsofPusingthehomogeneouscoordinates.So, AsubspaceofViscalledtotallyisotropic,if(v;v0)=0whenevervandv0arebothinthesubspace.WeletLbethesetoftotallyisotropic2-dimensionalsubspacesofV.Hence, Thetriple(P;L;I)withthenaturalrelationofincidencebetweenthepointsandlinesisanincidencestructure.FromnowonalltheincidencesystemshavetherelationofinclusionsowewilldroptheletterIfromthenotations. Notethateverylineof(P;L)hasq+1pointsinitandeverypointiscontainedinq+1linesfromL.Moreover,onecanseethatgivenanyline`andapointpnotonthatlinethereisauniquelinethatpassesthroughpandintersects`.Hencethisincidence 26

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19 ]. Fixapointp0=he0i2Pandaline`0=he0;e1i2L.Forapointp2P,wedenep?tobethesetofpointsonallthelinesthatpassthroughp.Thus, LetP1bethesetofpointsnotinp?0andL1bethesetoflineswhichdonotintersect`0.Henceotherincidencesystemsofinterestare(P1;L1),(P;L1)and(P1;L).LetM(P;L)betheincidencematrixwhoserowsareindexedbyP,andthecolumnsbyL.Similarly,wegettheincidencematrixM(P1;L1),whichcanbethoughtasasubmatrixofM(P;L).WecanreordertherowsandcolumnsofM(P;L)sothatthepointsinp?0comeontopandthelinesinLnL1comerst.So,wecanvisualizethetwoincidencematricesasfollows 3{14 )and(P1;L1)fromaboveareisomorphic. Proof. 22 ]. Hence,M(P1;L1)anditstransposeareparitycheckmatricesforLU(3;q)codes.SincetheLU(3;q)codeisbinary,wewanttoknowthe2-rankofthematrixM(P1;L1). 22 ],P.SinandQ.Xiangobtainedtheformulaforthecaseofoddq.Weusedierentmethodstoobtaintheformulaforthecaseofevenq.Wegivedetailedproofsfortheevencaseandstatethecorrespondingresultsoftheoddcasewheneverpossible. 27

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Wecantalkaboutasimilarorderingforthesetoflines.So,L=f`0;`1;:::;`q3+q2+qg: WedenotebyF2[P]thespaceofF2-valuedfunctionsonP.WecanthinkofelementsofF2[P]asq3+q2+q+1componentvectorswhoseentriesareindexedbythepointsofPsothatforanyfunctionf,thevalueofeachentryisthevalueoffatthecorrespondingpoint.Thatisf=(f(p0);f(p1);f(p2);:::) Thecharacteristicfunctionpforapointp2Pisthefunctionwhosevalueis1atp,andzeroatanyotherpoint.Thus,pistheq3+q2+q+1componentvectorwhoseentrythatcorrespondstopis1,andalltheotherentriesarezero.Hence,0=(1;0;0;0;:::)1=(0;1;0;0;:::)2=(0;0;1;0;:::):::etc:

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1 ]],andthesecondwasprovenin[Theorem1,[ 20 ]]. 2.Ifqiseven,the2-rankofM(P;L)is1+1+p 2!2t+1p 2!2t: 22 ]. 2(4{6) 2(4{7) Hereweprovethecorrespondingtheoremandcorollaryfortheevencase. 2!2t+1p 2!2t2t+1(4{8) 2!2t1p 2!2t(4{9) Forsimplicity,mostofthetimewewillnotmakeadistinctionbetweenthelinesandthecharacteristicfunctionsofthelines.Forexample,wesayasubspacespannedbylinesinsteadofcharacteristicfunctionsoflines. 29

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LinesthatareincidentwithXandY. LetC(P;L1)bethesubspaceofF2[P]spannedbythelinesofL1,C(P1;L1)denotethecodeof(P1;L1)viewedasasubspaceofF2[P1];andletC(P1;L)bethelargersubspaceofF2[P1]spannedbytherestrictionstoP1ofthecharacteristicfunctionsofalllinesofL.Thatis,ifM(P;L)isthematrixasin( 4{5 )wheretheblocksnamedasfollows B C D ThenC(P;L1)isthecolumnspaceof[BD],C(P1;L1)isthecolumnspaceof[D],andC(P1;L)isthecolumnspaceof[CD]. Weconsiderthenaturalprojectionmap givenbytherestrictionoffunctionstoP1.WedenoteitskernelbykerP1. LetXbethesetofcharacteristicfunctionsoftheq+1linespassingthroughp0,andletX0=Xn`0:Wealsopickqlinesthatintersect`0atqdistinctpointsexceptp0,andcallthesetoftheselinesasY.Inthegure 4-1 thesolidlinesrepresentthesetXwhilethedashedlinesrepresentachoiceofY. 30

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ItisimmediatethatthesetsX;Y,andZaredisjoint.Ifaline`isinX,allofitspointsareinp?0.Thus,itsimageunderP1isallzerovector.So,XkerP1:AlsowenotethatjX0[Yj=2q,whilejZj=dimF2C(P1;L1). Thefollowinglemmaandcorollarywereprovenin[ 22 ]. Proof. dimF2LU(3;q)=q3dimF2C(P1;L1)(4{13) BytheabovelemmaandthefactthatjX0[Y[Zj=2q+dimF2C(P1;L1),wehave dimF2C(P;L)2q+dimF2C(P1;L1)(4{14) Combiningthetwoinequalitieswegettheresult.

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Pickapointp2Pwhichisnotin`,`0,or`0.Byquadranglepropertiesthereisauniqueline0,throughpthatintersects`.Similarly,thereisauniqueline0,throughpthatintersects`0.Letp1denote0\`,andp2denote0\`0.Wecanseethisinthegure 4-2 Figure4-2. Lines0and0. Leta;b;c;e2Vbethegeneratorsofthepointsp1,p2,p,andprespectively.Thatis,thatp1=hai,p2=hbi,p=hciandp=hei.Withoutlossofgeneralitywecanassumethat(a;b)=(e;c)=1.Wecanwritethelines0and0intermsoftheirgeneratorsas0=ha;ciand0=hb;ci.Thus,thepointsof0otherthanp1areoftheformhc+aiwhere2Fq.Similarlythepointsof0otherthanp2areoftheformhc+biforsome2Fq. Througheverypointof0,thereisauniquelineintersecting`0.Beingin`0thesepointsareoftheformhb+eiforsome2Fqasingure 4-3 32

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Morelinesinthegrid. Sincetheform(;)isanalternatingbilinearform,andhc+ai,andhb+3iareonthesameline, 0=(c+a;b+e) (4{15) =(c;b)+(c;e)+(a;b)+(a;e) (4{16) =(c;e)+(a;b) (4{17) =+ (4{19) Thus=inFq.Thenfor2Fq,thelinethroughhc+aithatintersect`0is=hc+a;b+ei:(gure 4-4 )Similarly,wecanshowthatfor2Fq,thelinethrough Figure4-4. Linesofthegrid 33

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=fj2Fq;=hc+b;a+eig(4{20) =fj2Fq;=hc+a;b+eig(4{21) Notethatthelinesin(respectivelyin)donotintersecteachother. Nowwepicktwolinesandforsome;2Fq:Then,=hc+b;a+eiand=hc+a;b+ei.Wewanttoshowthatthesetwolineshaveanon-zerointersection(gure 4-5 ). Figure4-5. Intersectionofand. Hence,andintersectats,andbyquadranglepropertiesthisistheonlypointofintersection.Hence,everylineinintersectseverylinein.Thus,thereisagridoflinesbetween`and`0.Moreover,thelinesin[areinL1(gure 4-6 ). Figure4-6. Summinglinesinthegrid. 34

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19 ],section1.3,p.4].Whenqiseventhisquadrangleisknowntobeself-dual[[ 19 ],3.2.1].Hence,thelinesofW(q)areregularforthecaseofevenq.Thusonecanshowthatthereisagridoflinesbetween`and`0.Thismeanstherearetwosetsoflinesandsuchthateachsethasqelements,eachlineinintersects`nfpganddistinctlinesofintersects`nfpgindistinctpoints.Similarly,eachlineinintersects`0nfpganddistinctlinesofintersects`0nfpgindistinctpoints.Moreover,everylineofintersectseverylineof. Figure4-7. Gridlinesbetween`and`0. Weaddcharacteristicfunctionsoftheselinesandget

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Thusanyline`2LisinthespanofX0[Y[L1.ItremainstoshowthespanofX0[Y[L1isthesameasthespanofX0[Y[Z: LetR=k[x0;x1;x2;x3],betheringofpolynomialsinfourindeterminatesoverk.WecanthinkofanypolynomialinRasafunctionink[V]:Inordertondthevalueoff(x0;x1;x2;x3)2Ratv=(a0;a1;a2;a3)2Vwejustsubstitutexiwithaiforalli.Thus,thereisanhomomorphismfromRtok[V]thatmapseverypolynomialtoafunction.OnecanprovethatthishomomorphismisinfactanisomorphismbetweenR=Iandk[V],whereIistheidealgeneratedbyf(xq0x0);(xq1x1);(xq2x2);(xq3x3)g. 36

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Thus,foreachp=hvi2Pthevalueoffonpnf0gwillbeconstant.HencefcanbethoughtasafunctiononP.WecandothisbyprojectingfontotherepresentativesofpointsinP.Ontheotherhand,anyfunctionf2k[P]canbeextendedtoafunctionf2k[Vnf0g]kbydeningthevalueoff(v)tobethesameasf(p),wherepisthepointsothatv2p.Thus,thereisaonetoonecorrespondencebetweenk[P]andk[Vnf0g]k,andk[P]canbeembeddedintok[V]k. Sincek[V]'R=I,thereisaspaceRPwhichisisomorphictok[P],andthatcanbeembeddedinto(R=I)k.ElementsofRPareclassesofpolynomialsthatmaptok[V]kundertheisomorphismbetweenR=Iandk[V].LetRPRbethesetofrepresentativesofelementsofRP.Itisnotdiculttoseethatforanyelementg+IofRPtheuniquerepresentativeginRPwillbeahomogeneouspolynomialwhosetermshavedegreeswhicharemultiplesof(q1).Inthiscase,thesetofmonomialsoftheformxm00xm11xm22xm33in 37

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Foraline`2L,let`bethepolynomialinRPthatcorrespondstothecharacteristicfunction`of`ink[P].So, (1+(3Xi=0aixi)q1)(1+(3Xi=0bixi)q1)+I;(4{27) whereai;bi2ksuchthatthe2-dimensionalsubspaceofVgivenby 38

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4 ]. Proof. 4 ]withm=2andr=2. 39

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InthissectionwewillndthedimensionofC(P;L)\kerP1,whereP1:RP!RP1istheprojectionmap.ElementsofkerP1aretheclassesofpolynomialswhosevaluesatthepointsofP1arezero.Weknowthat Thus,anyelementoftheform(1+xq13)f+IinRPisinthekernel.Ontheotherhand,f+I=(xq13+1)f+Iforanyclassf+I2kerP1.Thisisbecauseforanypointp2P,thevalueof(xq13+1)fiszeroifp2P1,andf(p)otherwise. Proof. Fortherestofthesectionwexanelementr+IofkerP1\C(P;L).LetrbeitsuniquerepresentativeinRP.Sincer+Iisinthekernel,r=(1+xq13)h(x0;x1;x2)forsomeh2RP.Sincer+IisalsoinC(P;L),andC(P;L)isspannedbythecharacteristicfunctionsofthelines,thetermsofr+Ihavedegrees0;q1or2(q1).Wealsoknowbythelemma 18 thatr+Iisinthespanof.

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Sincer+IisinC(P;L),andxq13misamonomialofr,thedegreeofxq13misq1or2(q1).Sincemisnon-constant,deg(m)=q1:Hence, Since2t1isanoddnumber,k0=1.Thenweget andsok1=1.Werepeatthisprocessuntilwegetki=1foralli. Supposeoneofthemonomials,say[g0;:::;gt1],ofhhasx0init.Sogi=x0forsomei.Then, isamonomialinr.Weknowthatrisalinearcombinationoftheelementsof,so,rshouldalsocontainthemonomial [g0x3;g1x3;:::;gi1x3;x1x2;:::;gt1x3]:(4{35) Notethatthedegreeofx3inthismonomialisdierentfrom0orq1.Howeverthisisimpossiblesincer=xq13h+h,thedegreeofx3inanymonomialofrmustbeeither0 41

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4{35 ).Hence,hdoesnotcontaintheindeterminatex0. 22 ],theproofworksthesamefortheevencasealso. Proof. 15 appliedtop0,weseethatif`and`0areanytwolinesthroughp0otherthan`0,thefunction``0liesinC(P;L1).ItisalsoinkerP1.Thus,wecanndq1linearlyindependentfunctionsofthiskindasdescribedinthestatement.ThenthedimensionofkerP1\C(P;L1)isgreaterthanorequaltoq1.Ontheotherhand,sincenoneofthelinesinL1hasacommonpointwith`0,C(P;L1)isinthekerneloftherestrictionmapto`0,whiletheimageoftherestrictionofkerP1\C(P;L)to`0hasdimension2,spannedbytheimagesof`0andp0.Thus,kerP1\C(P;L1)hascodimensionatleast2inkerP1\C(P;L),whichhasdimensionq+1,byCorollary12.Hence, dim(kerP1\C(P;L1))q1:(4{36)

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Therefore,Z[X0[YspansC(P;L)asavectorspace.So, dim(C(P;L))dim(C(P1;L1))+2q(4{37) andthisimplies dimLU(3;q)=q3dim(C(P;L))+2q:(4{38) 43

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[1] B.Bagchi,A.Brouwer,H.Wilbrink,Notesonbinarycodesrelatedtotheo(5,q)generalizedquadrangleforoddq,GemonetriaeDedicata39(1991)339{355. [2] F.Buekenhout(ed.),HandbookofIncidenceGeometry,ElsevierScienceB.V.,TheNetherlands,1995. [3] P.Cameron,J.V.Lint,Graphs,CodesandDesigns,CambridgeUniversityPress,Cambridge,1980. [4] D.Chandler,P.Sin,Q.Xiang,Incidencemodulesforsymplecticspacesincharacteristictwo,arXiv:math/0801.439201(2008). [5] P.Dembowski,FiniteGeometries,Springer-Verlag,NewYork,1968. [6] J.E.F.Assmus,J.Key,DesignsandTheirCodes,CambridgeUniversityPress,Cambridge,1992. [7] R.Gallager,Low-densityparity-checkcodes,IRETrans.InformationTheoryIT-8(1962)21{28. [8] J.Goethals,P.Delsarte,Onaclassofmajority-logicdecodablecycliccodes,IEEETrans.InformationTheory14(1968)182{188. [9] R.Graham,F.MacWilliams,Onthenumberofinformationsymbolsindierence-setcycliccodes,BellSystemTech.J.45(1966)1057{1070. [10] N.Hamada,Therankoftheincidencematrixofpointsandd-atsinnitegeometries,J.Sci.HiroshimaUniv.Ser.A-I32(1968)381{396. [11] N.Hamada,Onthep-rankoftheincidencematrixofabalancedorpartiallybalancedincompleteblockdesignanditsapplicationstoerrorcorrectingcodes,HiroshimaMath.J.3(1973)153{226. [12] J.Hirschfeld,ProjectiveGeometriesOverFiniteFields,OxfordUniversityPress,NewYork,1979. [13] J.-L.Kim,U.Peled,I.Pereplitsa,V.Pless,S.Friedland,Explicitconstructionofldpccodeswithno4-cycles,IEEETrans.InformationTheory50(2004)2378{2388. [14] Y.Kuo,S.Lin,M.Fossorier,Low-densityparity-checkcodesbasedonnitegeometries:arediscoveryandnewresults,IEEETrans.InformationTheory47(2001)2711{2736. [15] F.Lazebnik,V.Ustimenko,Explicitconstructionofgraphswitharbitrarilylargegirthandofsize,DiscreteAppliedMath.60(1997)275{284. [16] F.Lazebnik,A.Woldar,Generalpropertiesofsomefamiliesofgraphsdenedbysystemsofequations,J.GraphTheory38(2001)65{86. 44

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F.MacWilliams,H.Mann,Onthep-rankofthedesignmatrixofadierenceset,Inform.andControl12(1968)474{489. [18] R.McEliece,TheTheoryofInformationandCoding,CambridgeUniversityPress,Cambridge,2002. [19] S.Payne,J.Thas,FiniteGeneralizedQuadrangles,PittmanAdvancedPublishingProgram,Boston,1984. [20] N.Sastry,P.Sin,Thecodeofaregulargeneralizedquadrangleofevenorder,Proc.SymposiainPureMathematics63(1998)485{496. [21] P.Sin,Thep-rankoftheincidencematrixofintersectinglinearsubspaces,DesignsCodesandCryptography31(2004)213{220. [22] P.Sin,Q.Xiang,Onthedimensionsofcertainldpccodesbasedonq-regularbipartitegraphs,IEEETrans.InformationTheory52(8)(2006)3735{3737. [23] K.Smith,Onthep-rankoftheincidencematrixofpointsandhyperplanesinaniteprojectivegeometry,JournalofCombinatorialTheory7(1969)122{129. [24] R.Tanner,Arecursiveapproachtolowcomplexitycodes,IEEETrans.InformationTheoryIT-27(1981)533{547. [25] J.Tits,Surlatrialiteetcertainsgroupesquis'endeduisent,Inst.HautesEtudesSci.Publ.Math.2(1959)14{60. 45

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OgulArslanwasbornin1975inTurkey.ShereceivedaBachelorofSciencedegreeinmathematicseducationin1997fromMiddleEastTechnicalUniversityinAnkara.Aftergraduation,OgulworkedasahighschoolmathematicsteacherinAnkaraforseveralyears.ShestartedgraduateschoolatWayneStateUniversityin2002andreceivedaMasterofArtsdegreeinmathematicsin2004.Inthefallof2004,OgulstartedgraduateschoolatUniversityofFlorida.ShewasawardedaPh.D.degreeinmathematicsfromUniversityofFloridainAugust2009. 46