# Fingerprints of Projective Planes of Order 25

Here are the fingerprints of the 193 projective planes of order 25 which are known to me. This isomorphism invariant succeeds in distinguishing all non-isomorphic pairs among the known planes of order 25, except for the Foulser planes b1 and b2 (and their duals b1Dual and b2Dual). These pairs of planes are, however, distinguished by other invariants, such as the 5-rank.

## Definition of Fingerprint

Given a projective plane of order n (which has N=n2+n+1 points and N lines), consider the N-by-N matrix A with entries aP,L defined as follows. For each point P, choose an arbitrary but fixed ordering of the lines through P as 1st, 2nd, 3rd, ..., (n+1)st. For each line L, choose an arbitrary but fixed ordering of the points on L as 1st, 2nd, 3rd, ..., (n+1)st. Now for each non-incident point-line pair (P,L), the incidence relation naturally provides a bijection between the n+1 lines through P and the n+1 points on L. Relative to the chosen orderings, this gives a permutation of degree n+1. Define aP,L to be the sign (1 or −1) of this permutation. If P lies on L, define aP,L=0.

Note that different choices of orderings for the points and lines have the effect of possibly permuting the rows and columns of A, and possibly multiplying certain rows and columns by −1. Therefore the multiset consisting of the absolute values of the entries of the matrix B=AAT is an isomorphism invariant of the projective plane. This multiset (i.e. set with multiplicities attached to its elements), due to J.H. Conway, is known as the (complete) fingerprint of the plane.

For translation planes one obtains a computationally cheaper isomorphism invariant, called the (restricted) fingerprint, by replacing A by the (N+1)-by-(N+1) submatrix thereof having rows indexed by the points on the translation line, and columns indexed by lines through the origin. In the case of translation planes of order q2 with kernel containing GF(q), this restricted fingerprint is more readily computed directly from the corresponding ovoid of the Klein quadric, as shown by G.E. Moorhouse, Two-graphs and skew two-graphs in finite geometries, Linear Algebra and its Applications 226-228 (1995), 529-551.

Since I am comparing more planes than just translation planes, I have listed only the complete fingerprints. The restricted fingerprints of the translation planes of order 25 have been published by C. Charnes, Quadratic matrices and the translation planes of order 52, in Coding Theory, Design Theory, Group Theory (Proceedings of the Marshall Hall Conference), ed. D. Jungnickel and S. A. Vanstone, J. Wiley & Sons, 1993, pp.155-161. Note however that Charnes there uses a slightly different nomenclature for the 21 translation planes than that which appears in the final corrected and published version of the paper of Czerwinski and Oakden.

## Table of Fingerprints of the Known Planes of Order 25

No.

Plane

Description

Fingerprint

1*

Desarguesian

0423150625651

2

Hering

5641250057215000058011250058411250060035650625651

Dual Hering

10042016825000200180300203607500600390030625651

3

Hall

57215000057612500580750005841500005921125060024400625651

s2Dual

Dual Hall

0202425000100360200240400185047500600390012625651

4

Walker

5641250005683125057631250580125000584312505926250060016900625651

a3Dual

Dual Walker

0502431250100200200100300200400100600391250625651

5

Dickson nearfield

5681500057612000058027000060018150625651

Dual Dickson nearfield

019210043216830000200245522500600390002625651

6

Rao

017065042025008225001222500163750241250625651

Dual Rao

0422746100324200363003640066002625651

7

Czerwinski & Oakden

0100006015000064900006830000721050001681250196300002006250600650625651

Dual Czerwinski & Oakden

014485000403000064120000881500096130000100336104120000184125020016821612506002625651

8

subregular

025815086000012600001645000625651

Dual subregular

0422886200963009640072625651

9

Foulser

0656504260000897500625651

Dual Foulser

0422578100416200156625651

10

Czerwinski & Oakden

025815041200008150001230000625651

Dual Czerwinski & Oakden

04228861001922002430048625651

11

subregular

07200642250072112500764500080112501961800002004500588500059215000596150006005200625651

Dual subregular

036100288120100002001802161200002241800002322250026490000300804004250024625651

12

Czerwinski & Oakden

5683000057245000576127500580900005848250058850005922625060016900625651

Dual Czerwinski & Oakden

0204241500010021620018030083121750040042600390000625651

13

Rao

048005615000643000068150007215000767500192750001969000020033000204150005682500576437505804250058415000588100006009100625651

Dual Rao

0238100236120500013615000200124216500024875003005234421000035212000039260000625651

14

subregular

0198150412000081500012600001630000625651

Dual subregular

0422790100192200243009640048625651

15

Rao

01130004150000837500121000002003000240375026015000400300600600625651

Dual Rao

02516040750006430000080750010026412015000200603001604006625651

16

Walker

564200000584500005881000005965000060023150625651

Dual Walker

12020000200802161250030048050080600390010625651

17

Czerwinski & Oakden

024005637506815000192600001969000020015005602000056880000572200005804500058432500588300005961000060013000625651

Dual Czerwinski & Oakden

0962420000100272120750020018626450003004840032472300000480600005001652030000625651

18

Czerwinski & Oakden

016700437065081500012106001610200625651

Dual Czerwinski & Oakden

042252410055220036300324006625651

19

Foulser

0656504260000897500625651

Dual Foulser

0422578100416200156625651

20

Czerwinski & Oakden

0178150413000088500012200002010000625651

Dual Czerwinski & Oakden

04227581002082001363003250016625651

21

013250415000012125000205000241125004013750160125024012506001150625651

0125086250016100000205000245000032156250401250486250562500010024012015000300200600210625651

22

402500601000805009220000961000010020700112900014010001642000016830000180500018820000196250002002502041000021620000220415002284000023620000240130002442000026015000268350002722000027610000280550028450005202506002950625651

8500012250056100006020064360007230500801000084200009650001004010420000108100001203000140200014810000180250018825001922500220260002241250022830000232250024420000248262502525000260250026412502725000276150002801000284250002882750029620000300200308250032850003321000034020003565003605003761000380300038440004003000600210625651

23

Czerwinski & Oakden

01381504120000812000012225001622500625651

Dual Czerwinski & Oakden

04226941001922001923003640036625651

24

42400086000126000161500020183002860003610004015005212000566000609006415000681800072700080330088150096120001002401043000108200011239001201050132560013612000140100501441800014827000152150015620500160219001643000168300017221000176210001806000184240001889000192300019675002003002041000020860002129000216300022040502285500240285028090031210003409003485006001410625651

030008300016300020150024600028300036150040135004815005210500563000060600064150006815000721125076135008090008422508814250921350096310001002340104150010812000112975012424000128120001323000136225014073501444500148630015230001608100164675016845017275001767501801200188825019275019690002005250204750208300021630002201800022417250228900023615024060024430025610502604350264315026860028015002846600288390030012503041503123003169003402400360600380300400600600210625651

25

1220002036004036005640006812000723600076240008024000884800096120001003000108240001163600012048000124480001284800020050021625000220720024018003005040320480042012006001410625651

36600056600064360007224000764800084540008812000926000100120112120001166000120264001282400013224000140324001442700015224000160240019660002004980220600023224000236600024036003003203121000600930625651

26*

(Ordinary) Hughes

0437104372000407440625651

27

8216000122400024200028120003648000441200052480005624000721000847000120540014012001804000216500220240030014403126000320480040820006001410625651

82400020300004024000441200048960005630006061200642400072700092108000962400010024017260003002003603004402400600810625651

28*

0190820000163500020200247000028400003280004010325044650004820000527000561000064100084300088125009650001004000108500012020001606000200225022010002802503001200392500400100600710625651

29

0401620000202003230004060005220006030008040009210009640000100302001044000011210000116250001203025012820000132450001363250014020001442000014830000152100001601000168100001722500019225020073002201000240150032010003401000400300600610625651

02000042500081000016250002050002845000326000040200044400005650006820007210008011050084300008870009240001005240108100014020001602000180200020021003002003881000400104082505602500580100006003350625651

30

02404180001224001648000201704002410002836000322400401236006024008840001201200140120014824001606001881000232240038820003922000576500592450600960625651

054020412000824000124800024910002826400324800044240048120005672006012000641200686000764800080240001001201402402002403843000400184402400456300600612625651

31*

Exceptional Hughes

02480283720007231000100868020055804002480600930625651

32

01275041800081200012196001624000203240024335003239000363400040120004412000482500052150005618000603090064150006860007220007627000843000886600966000108600011230001201750140600160150168130024060030030032015040018604126004206005929006001590625651

01539042400081500122700016127502019500241900028360003260003622500401500044135004830056510006015600643000682520072150080945084300092600010060001043001082400011260011612001209600124720012812600136315014018001443600152120001563000160240017215001802400200300208150022012002801200300200600210625651

33

01218042400082000162400020420024100032120036600040150060120006424000722400076900008060008410800088240009210001002000104600010812000116240001203001321200140300015240001686001923003006003202100400720528150576500600600625651

020424000163030020120241000366000403000442400048240005224000563600064600006860007230000806000883000092300010012010415000116360001206600128600132180001407800144300014812001566000160810016815001803000192120020024022013202403600280120030012004001860012625651

34

3232004061200449000483200052520005610006032000648000683200728000763400080160008440008832400961600010017440112160001161600012048001403680015280001564000180400200200216250022080024040024816002608006001410625651

163400020160288000326400036100004420005610006080806412000724100076120008024000844800088320009216000104800010840001164800012021200156400016032001724000188400200400204800208160021216002204000224240023640002401600280500300200360800600210625651

35

050041200081200016120002024000241200028120003216200361800040144004424000481860052376005648000602400644200068264007219000762400080120088216001001240104100010810001121200120350012820001326600140240014818001521000160600168120020045024060040060600600625651

0610044200012270001660000203300024250002830000324500361200040312004418000486000561500601200064300068150008076208490008830009212600100801041740010812000112600120720124300013660014012001444200148120001561200160120016460018060002002560300124060030625651

36*

02951042000085950012780001636000204600024225002820000322200036900040924044130004812000524000561000060304064800080200885009220001001601044500108800011210001201050140200018810002002402401002805030040400400600120625651

37

2806003004803121200340240034824003605036850380240038834003962400040096420480042448004362400045630046410048440004889600049630050410055210005721000576125005807500058415000060012174625651

02010036020024023224000264100040018456300460480046424005449600054819200055248000556120005682400057248005761200584600058848006001212625651

38

044780470000844000123880016200002020000242200028200003230000364200040134004440004812800526400569000606400685600722000803001001000112400120300128200132120014014001524001602200168600192100200320220800232400240400248400260200312400320200528150600600625651

024520440000817200121240016246002027000244900028504003219800361240040156004432800481800052820056128006014640641140068200764800804200842000884000964001009000104400120140014088016016001804020018402808003008004001860012625651

39

037060445000867000126390016510002099002424000281600032102003639000401420048132005235005670006024006815008090088130010060104100011260012016501361000140280014830019250019650020051021215002206002401502683003003003121200320150392300400304203004321504401505001260600780625651

021020493008138001212300162430020243002426500282700032900036483004024600441410048123005224300601410064100506815007245007620700802070084177008812000961650100342010415010830001121500116300012015901321500136225014063001606001641500176750180120020023404001860012625651

40

0340804290008200001248200163600020292002430000283300032426003630000401920044100004814600522000564000601580068600762000804600842000104200011220012010001285013260014028001481000160120016840019280020028021214002201000232170024040024820031220032040040040600600625651

0100204492008450001236000163150020250002430000281400032800036226004026004430200482000052840056160006054006470006844007211000762600801020084980088250092460010060010812001122000116140012021601244001322001408001442001561000160280016850018040018410020024022040024017002802003004003804004001860012625651

41

82000164000202000024360002860000324320036370004029600441400048820052248005620000609000643400068106007210500762000804000845000882400924001008200104100001081100011240011610001206700128200132800140220016840019220020030021216002201200232400240500280100300240320400600610625651

01600041000081800012100001637000202100024192002816000321000363600040340004460004817000521800056280006026000641200068600072410076140008014000844000886000921000968001001452010432001122000116200012034001241600132400136300014076401444001484001564001608801641200172400180840184500196200200186020850021620024012002608003008040041660014625651

42

070480008810012128001615000208400241600028240003250000362100040276504428000482080052164005626000603100068780072180008082008480008810000921000961000100808010410001082000112440011620001204450124100013232001365001405600148680015240001601400164400016820020034020420002121200220120023220030080312120040022044020052050600610625651

0100004200008310001212400163620020240002442400282280032338003610400403490048140005212000562620060104006450068400072210076400802000849200883000921400968000100332010460001082000112121001201001242000128410014024401441000152205015640016030401644400176400180800184400192502001140220800236400240160024440030080400164481000456250512500520750600514625651

43

02400044400082200012284001636000202140024850028170003296003632000401600044270004850200524000569000602180064800076100084120008840001004080108200011240001203350132120014015001602001684001924002009102162000220140029280030076031226003204005001006001150625651

02000414400870001218500161340020558024180028340003214800361940040241604440200482600052340005618250601020064360006836000722200076700080560084200092550010040001082000112140011620001203000124100013210013650016020016450017620001808001882000200115022020002401200300200308400328100340100440100520200600410625651

44

890001292001618000208400245002832000323440036520004011200442400048292005215000562400060132006480006811200724000761800080256008814400921200010016010412000108800011280012010600136200014010000160400200200212200022040024040024840028010030048031240032040039240040040600670625651

08000410000816000123600016230002016080242400028140003210000366000401908044400048200052180005625000608000641820068200007210000761800080208084200008816000968000100200010418400108400011212000116800012056001241200136400140400014430001484001561200160440016812001764001801600200800220800248100260800280100030020040016600194625651

45

022060420000845000121280016150002028600244150028220003234400361300040190004480004811800521640056180006023100641400068800726000764000801140088150092850096400010051201122400120105012850013254001402600148200152200160400168600200180212800220200232200240110030040312200320200340200600700625651

02514044800081720012170001624000203100024280002829400321150036190004019740441020048920052520056130006023000649100687000722300076520080700084240088770092400961900100236010415001081200112400011626001201580124200013610001401600144300160200016410001681700180400188200200156021650022040023220024010002604002809003004040041660014625651

46

026140459000854050124870016390002019600242200028150003226200361500040263004440004811900524500567000608250647000683400722500762000801000841000881300921000961000100122010410001082500112800120270012450013280014017001486001603501684001801502001602124002203502325002409502482003001003205033210034050392504002044050600610625651

01108044850084070012358501630300203762024324002815750323335036226004023490441760048460052109505695006079006463006847507216007637008023008420008835092100096110010029001045501081800116155012027001402250152550160850168100172200176100180600188100196300200300220800240600260200280200300600320200340200600210625651

47

01000043800082000012312001632000201680024340002833000323530036170004018500447000488600521820056230006019000642050068130008010008440008866009220010040801042000112400120200132800140140014820016060016840019245020073021240022012002404003004031240032050034020039220040020420600440200592100600730625651

0502041340082720012240001629000203500024252002810000324000362500040150004411600484160052142005611150601820064206006814600724007644008065808450008811000921500096150010019601045050112640011620012086001281000136900140248014416001601600164200176400180124020012402208002404002604004001860012625651

48

0857043300084450012457001636000204490024355002833000323590036210004010800441000048100005211900568000607000646000686007220007620008012008420009210001001001127001201600124100013250014024501485001522001603501682501801501923002004502129002208002325002404002482002601002681003201003401003921004001044050600670625651

0195704359508407501227800164005020398502438650281160032225503619700401555044775048405052945056103006012000641500068560072435076235080514084785088365092590096200100260010470010825011260011615001202300128750132150136600140170014810015610016019201725001765001807002005402204002406002806003004004001860012625651

49

0220904460008410001237600162600020349502443500283500032287003627000401440044500048158005284005660006051506430006843007250076500080200841000881100921001003001122000120900132800140145016030016810019225020029021210002201100232400240400248200280502921003123003401003921004002042050500240520200600710625651

09020433200834750123130016314502027250242635028163503217000362550040289004412550481225052157005620350605550641590068710072710076495080399084720088365092700962400100422010428001081200112100116900120348012455013210013645014023201446001521001561000160800180180020013402202002404002801003002004001860012625651

50

0130404290008240001238500162200020177002428500282650032235003647000402765044250004865005212100568000601665064500068760072700076800080320084700088200962000100126010830001122300116500120130013290013610001402100148300160500168450192100200702121100220850240450248400320150400170600610625651

01502044530082725012331001627500203250024352502814900322250036199504017700441750048135505214700561240060815064110506880007240007620508039008461008835009260096370010025201043001081501121100116200120224012455012823501322001365001401450148200152501562001603700180170018450019220020079021650022010002361002408002443002561002641002804003042503202003402003601004001860012625651

51

015570428500840000123775016335002041300244050028320003225100362200040157504495004811450521535056950060118506445006876507215007610008021008415008816009210009610001005801081000112800120125013265014022001483001603001684502002502129002207002324002403502485026050300130312200320350340504006050060600600625651

018520432500831800123415016252502030650244245028223503227300362430040185504420150489000527950567800601230064112006857507245007656008061508433008810509295096800100522010495010830011225011640012012101247001281001323001363501401380160120016425016810017610018011001845019620020010402085022013002242002281002321502401000252502602502801003003003401004001860012625651

52

03503046900085300012367001627000203835024290002822500322095036150004013750448500487850527850565000605200642500682950727000765008014008870092509650010040011217001201650132800140165014850152100160500168150180100192250200390212135022075023225024015025210026010029250300110312250320250340503925040060412100420200500530600780625651

0131204255008213001218700162490020265002429050282710032256003625850402470044283004810050522030056182506012550641290068550072675076450080715084815088455092260096160010024501041350108105011225011635012017801241250128250132250136500140224015670016012401643501807002007402082502201002282502409002805003001003201004001860012625651

53

203050028200003230004015005620000606500681200076250008846500921000096600001004020010810000112100001161000012039750124150001285000136100001405500144200001522000160575016410000200300220100024050026020006001150625651

01129081000012150001650002025200242500361000040205004440000482100052500060325006425006840007212500761250080220009620000100500011215000116500012030000124200001401000156200001602500172125001841000018850002007000204250020850002401000244500248500260200026415002721000276100028012503002200600210625651

54

0808120001288002060028120003230003640004019600444200048242005212600562600060170006420000683720072800076200008016100841000088600092132009635000100141001041000108130001121420011670001205100124200012830001323600140380014822001522200160350168400192200200140204100021260022030023220024040030020312100320200600660625651

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85

0663041050081700012199001617000206950248000281150032270503621000401895044280004825750521895056200006018650641800068229007219500761800080121508460008810250924500967500100961010435001121550120195012410001286001321150136500140290014885015255016055016830017250020050212300220450228502325502404002483502605030010320100340100600600625651

0907042265082370012161001628400202300024241502826100321925036222004020050441525048125505212600567500601280064153506810750721150076121008061508410400885750923250965300100926010458501081550112375011680012041301242300128155013213501361350140208014420014835015250156300160144016420016820017235017640018015001844001882001922001961002002340204100208200212502164502208502245022810023250236100240120025215025650260400264100280600284503203003601004001848010060012625651

86

01263042250082350012219001628500202900024230002823050322585036170004022700442150048176005213900561850060234506415500681330072700076100008061008450008831009230009620001001600108500112550120165012450012850132110014024501486001522001606501682001921002001102124002205002321502408502483002605029210030060312503201503325034050600600625651

0195204285008255001225050162275020335002425300282670032222003625650401970044171004813350521600056980060610064124006882507260507654008099308446008844009245009628501005310104155010815501121100116135012019001247001281150132100136200140194014450014810015235015610016023801643501685017210018017501882001962002001890204200208502121002208002401200244100260100272502803003202003601004001860012625651

87

025030432500830500123655016395002036100242950028255003226000362150040194004414000481905052780056110006090006435006870507250007610008036508415008811009221009615001001130104500112115012020501321350140180014830015250156500160600168400180501925020010021260022080023230024050024850260502685029250300100312150320300340504008042010050010600600625651

0160204324008342501233600162965020306002436450283170032154503615300401855044218504811250521635056825060730064775068115507251507669508046508438008810092200096145010040201048001081001126001168001201710124205013250136250140203014430014810015250160220016825017215017650180800184100192250196100200114020410020850212502209502281502404002521002602502803003202003403004001860012625651

88

029030440500840500124165016365002032500242100028201003224450362150040168504414500481135052112505665006063006455006810100722500763500805100845008839009216009620001001301041500112900116500120145013219501401750148350152501605501681001805020010021214502209002322002406502481002605026850300100312100320150340504008042010050010600600625651

0173204432508360001231150162350020289502435300282810032237003627850401915044143004818000528800564900606250647600685750722050767050804600843150887509223009614001003920104275010830011230011612001202160124150012810013230013610014098014450015645016020001642001805001885019610020064020435021230022014002401002605502685027650280300284100288503002003045036010040011852020060012625651

89

02456043300083900012363501634500204495024270002824550322345036190004022350442300048167505284005675006010100643500685850723500802750885509265096500100100112650120125013213501402550148550152100160950168100180501921502001302125502207002321502406002481502601003001603122503201003325034850600600625651

016820438550837200123510016330002027550243440028266003222400362165040193504413550489750529150561415060111006411050683700724300761500804850843400881300921250961800100342010419501081001124501161100120316012430012810013250136400140148014450148501522501565016011501641001681001721501801400188501961502001740204502085021210021650220500236502404002601002802003003003401004001860012625651

90

017620428500839500122285016340002040500243650028325503226800361900040211004420000481655052100005616000601055064500068810072155076150080215088150925501001301045001129001201500128500132650136500140175014820015215016055016835019250200250212105022050023240024075024850260100300230312200320150332504001050060600600625651

01732043830082975012329501631550202935024384002821700321880036278504019000442245048830052835056840060795064102506839007232507630008048008445508827509229509642001006120104375010830011620012017601244001285001325013625014019801441001482501565501601200164500168501761001801600184100188501965020010402208002245023250240600244502602002801003002003401004001860012625651

91

015030440000825500123705016330002027300242700028355003235050362150040249504420000482030052125505611000606750643000685600723500762500803150841000886009250010010011275012014001327501402150148550152501606501681502002002121300220750228502322502402502482502605030025031250320150332504006050010600600625651

08270438100839850123235016294002032800243345028259503221000362760040137004419950486600521335056750060120506482506826007247007683008054008471008820009225009680010036201042750108500112350116100012018801283501322501401610144100148501521001562501601450164500168501721001765018012001841001885019210020013402209002401002603002801003002003401003801004001860012625651

92

07710421000812000129600169000202730024270002821000322640036230004024000441950048486005227300561800060148006460006810200729000761200080150848000883009260009610001006010430001128700120655013245001404050156500160900168300200240212300220195024075025230026030030036031230032030034030040030600600625651

0450043450084650012750016375002017250241275028240003238250366000401287044127504813500526300561950060104706445006821000721435076600080135008475008815009260009633001001800104240010815001124200116465012037501281350132750136900140315014415001483000156450160600018012001841502001500220600240600244150264450272300300200360300440300600210625651

93

01204041500081910012112001650002033000242700028240003225700362500040287504423000483260052640056280006017300642700068228007240007630008012250843000881000922009640001002801082000112800120145013210001401400148200160150018020019230020025021220022060023210024010025650030014031260032020034010039210040013044050600610625651

0121004600082550012290001621050202450024311502819000322150036297004015780448600482650052245005610400601440064495068230007216900769000802590847700888000963400100580010410001087001122000120170012810001322001361001402600148200152200156500160320016418001683501724001765501808001842001961002002260208100220400228200240800280600300403601004001660014625651

94

0261504620008505001234250162950020297502427500282500032212003619000401895044165004810200528700565000605950644500685500724500762000801008420008825509250100120112950120100013224001402650148600152150160550168300192100200190212250220650232100240500260502925030010312100320300332504002042010050010600600625651

0150004473008408501225800164380020355002431350282530032204503618600401825044208004858005275505676506012400647450683300726000764100802650848008815509210509630501003770104800108150112550116400120110012480012830013215013615014013401522001601390164250172600180104018430020086022011402402002605002803003001803402003801004001660014625651

95

011110429000840550124705016315002025150242400028285503234150361950040274504423000481250052100505611000601305064750068485072300076100080265084300088509210501005701045001121550116500120105013211501401700148450152250160750168350180250192100200150212400220750232200240300248502605030018031215032015033210039250400404205044050600600625651

015900435500837550123750016331002023050242785028290003224850362010040207904415300481585052890056112006011240647700686300724600763950806100841950883100928509615010021401041800108285011275011645012019001241501285001324001401450152100156200160185016420017215017625018070018450192501965020010602045021650220160022850236150240800244100252502605002801003001403401004001660014625651

96

019090441000852550123100016295002034950243650028205503225450361150040153004485004814300529850561300060805064950068450072605076200080555084200088370092150962000100150104100010810001122900120180013217501402750148300152100160250168500200270212650220750228502323502406002481002525026050292100300703125032025034850400104125044050600610625651

012750435350833350123245016248502031100243695028260003228400361780040269004419450487800521260056117506011410647750683400726400764400804380841700882750923500966001003500104500108600112850116160012020001244501282501321001402250152400160800164350172150180600184501963002008002043002209002405002601000280300300500340100600210625651

97

01558042450082250012361001633500203005024180002823000322515036205004017100441100048227505213350561300060108506410500681355072110007675008084008430008841509215009615001002640104100010830001122250116100012021001241000128150013225001402200148200152100156500160550168250192100200380212800220900224500232502404002681002885030090320250348504205050010600600625651

017190434500830150122575016277502037650241450028144003225000362100040263004423000481570052120005612600601050064885068755072680076300080465084905088105092270096270010047501041150108750112115011610501202250124190012875013255013623001402450144400156100160150016410016810017250018070018825019635020016002041502204002241502321002401400244100252502604002641002725028040028450300200340200360100400100600210625651

98

Wyoming

0960041280084000012800016288002018400241820028128003232000362880040289604412600481920052144005621000602512064480068800072800076800802464088410092640096960010016011216001168000120332012416001363200140104015240016024001804801844002004402081600216100280400320320344400520160600110625651

Dual Wyoming

014400464008208001242400161280020969624320002812800323120036280004013064441360048160052960056318006014416641920068224007216200761080080646484960088960092160009632001008001042800108520011216001241200156800172800200242648002801003007240012850012853620055240060058625651

99

0400004111600820400124000016768002024480242040028192003238400364800401056044480048320056150602400726007624001009601682001802002001140216100300240600120625651

03936046240086240012111600242400028684003219200361200404800444800561200609792684800722400922400100192120200136330015260020078300246004625651

/ revised September, 2006