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Some Algebraic Problems From Coding Theory

Permanent Link: http://ufdc.ufl.edu/UFE0024938/00001

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Title: Some Algebraic Problems From Coding Theory
Physical Description: 1 online resource (46 p.)
Language: english
Creator: Arslan, Ogul
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: coding, finite, generalized, geometry, incidence, ldpc, lumq, matrix, prank, quadrangle, representation, symplectic, theory
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

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.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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.
Statement of Responsibility: by Ogul Arslan.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Sin, Peter.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024938:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024938/00001

Material Information

Title: Some Algebraic Problems From Coding Theory
Physical Description: 1 online resource (46 p.)
Language: english
Creator: Arslan, Ogul
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: coding, finite, generalized, geometry, incidence, ldpc, lumq, matrix, prank, quadrangle, representation, symplectic, theory
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

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.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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.
Statement of Responsibility: by Ogul Arslan.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Sin, Peter.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024938:00001


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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