÷ƒ’À;è TeX output 2001.09.24:1631‹ÿÿÿÿ øÒ ýHÐ! ™/ß þ™‘rúúó¥!¢N ecbx1200»Some–¸new“triplanes“of“order“t• w“elv“eŽŸCç’¸Û¿ó 1ê± ecrm1000¹Sanja‘U Ruk‘ÿqÐa¸èvinaޤ™œ’½qDepartmen¸èt–U of“mathematicsŽ¡’—ùF‘ÿ*¸acultš¸èy–U of“philosoph˜y“in“Rijek‘ÿqÐaŽ¡’ˆ[9Omladinsk›ÿqÐa–U 14,“51000“Rijek˜a,“CroatiaŽ¡’–H“(e-mail:‘q€sanjar@map•Gef.p“efri.hr)ŽŸNk‘ó½HЃ ecti1000½R¾˜unning‘“°he‘ÿ}/ad‘w:‘pó  b> cmmi10µZŸÿóÙ“ Rcmr7±6Ž‘Ñ“¹acting–U on“2-(71,15,3)“designs“with“one“xed“pGoin¸ètŽŸ)ÌϒƸó‹–uÌ ecbx0900¿AbstractŽŸyñ‘&ß$óÙ.œŒ ecrm0900¾Up–NHto“isomorphism“there“are“72“symmetric“(71,15,3)“designs“admitting“an“automorphism“ofޤÌÏ‘order–bˆ6“acting“with“one“xed“pAÃoin¾Knoš¸èwn“non˜trivial“triplanes“ha˜v˜e“parameters“(11,6,3),–Z(15,7,3),“(25,9,3),“(31,10,3),Ž¡(45,12,3)– or“(71,15,3).‘XÖA¸èccording“to“[6],›íclassications“of“triplanes“of“orders“three,˜four,˜six“and“sev¸èenŽ¡are–Ü¥already“completed“and“there“are“2893“triplanes“of“order“nine“knoš¸èwn.‘T‘ÿ*¸riplanes“of“order“t˜w˜elv˜eŽ¡are–Ã>the“biggest“knoš¸èwn“triplanes.‘»ÙThere“are“11“symmetric“(71,15,3)“designs“kno˜wn“(see“[3],–ÞÅ[4],“[6]).Ž¡The–U orders“of“their“full“automorphism“groups“are“48,“168“and“336.Ž¡‘Let‘0¸D‘5²=‘Ç(¸P‘Ò}µ;‘ª¨¸BŽ‘/~üµ;›ª¨I‘Èâ²)–0¹bGe“a“symmetric“²(µv[Ù;˜kP—;˜²)“¹design“and“µG–Ǹ“µAut¸DG¹.‘e&Group–0µG“¹has“the“same“n•¸èum“bGerŽ¡of–ï¦pšGoin¸èt“and“blo˜cš¸èk“orbits.‘ALet“us“denote“the“n˜um˜bGer“of“µG¸¹orbits“b˜y“µt¹,‘GpGoin˜t“orbits“b˜y“¸PŸÿ±1Ž‘|sµ;–ª¨:“:“:Ž‘ÿ÷;‘ª¨¸PŸÿó 0e—rcmmi7´tŽ‘…V¹,Ž¡bloGcš¸èk– ]orbits“b˜y“¸BŸÿ±1Ž‘|sµ;–ª¨:“:“:Ž‘ÿ÷;‘ª¨¸BŸÿ´tŽ‘…V¹,‘Qand“put“¸jPŸÿ´rŽ›m¢¸j–Dz=“µ!Ÿÿ´rŽ˜¹,‘Q¸jBŸÿ´iŽ›TL¸j“²=“ Ÿÿ´iŽ˜¹.‘X•F‘ÿ*¸urther,‘Qdenote– ]bš¸èy“µ Ÿÿ´irŽ‘ LK¹the“n˜um˜bšGer“of“p˜oin¸ètsŽ¡of–—¬¸PŸÿ´rŽ‘N¹whicš¸èh“are“inciden˜t“with“the“represen˜tativ˜e“of“the“bloGc˜k“orbit“¸BŸÿ´iŽ‘TL¹.‘2ZF‘ÿ*¸or“these“n˜um˜bGers“the“follo˜wingŽ¡equalities‘U hold:ޤ™œŸóý’¨6ôtŽŸ €’¢€‰Ÿöüóú±u cmex10«XŽŽŸ ­†’¢²›´r7±=1ŽŽ’²œúµ Ÿÿ´irŽŽŽ’É L²=ŽŽ’ÚÓjµk‘û?;ŽŽ’©eʹ(1)ŽŽŽŸ"{Ÿóý’´tŽŸ €’‡XáŸöü«XŽŽŸ ­†’‡Šó´r7±=1ŽŽŸù<$’˜¨…² Ÿÿ´jŽŽ’˜¨…Ÿw‰fe oŸ (Öd:µ!Ÿÿ´rŽŽŽŽŽ’¥KHµ Ÿÿ´irŽ‘Aîµ Ÿÿ´jgrŽŽŽ’É L²=ŽŽ’ÚÓjµ² Ÿÿ´jŽ‘oŒ²+–8ൟÿ´ijŽ‘ Cظ“²(µk‘‰w¸“µ²)‘ª¨µ:ŽŽ’©eʹ(2)ŽŽŽ¡‘The›=­²(µt– ÷¸“µt²)¹-matrix˜²(µ Ÿÿ´irŽ‘Aî²)˜¹with˜en¸ètries˜satisfying˜propGerties˜²(1)˜¹and˜²(2)˜¹is˜called˜the˜orbit˜structureޤ™œfor–U parameters“²(µv[Ù;–ª¨kP—;“²)–U ¹and“orbit“distribution“²(µ!Ÿÿ±1Ž›|sµ;–ª¨:“:“:Ž‘ÿ÷;‘ª¨!Ÿÿ´tŽ‘…V²)¹,“²( Ÿÿ±1Ž˜µ;–ª¨:“:“:Ž‘ÿ÷;‘ª¨² Ÿÿ´tŽ‘…V²)¹.ŽŸ/w@Ç2Ž‘Áó·ág£ff cmmi12ÉZŸ(õ²6Ž‘ Ç]Çacting–G\on“a“triplane“(71,15,3)ަ¹F‘ÿ*¸rom–U noš¸èw“on“w˜e“shall“denote“b˜y“µG“¹an“abGelian“group“isomorphic“to“a“cyclic“group“of“order“6.Ž¡‘The–Ÿörst“step“in“the“construction“of“all“symmetric“designs“admitting“an“action“of“µG“¹is“to“determineŽ¡all–ùãpšGossible“orbit“distributions“and“to“nd“all“p˜ossible“orbit“structures“related“to“them.‘SThe“follo¸èwingŽ¡facts–U are“useful“(see“[5]):ŽŸ™œó]fŒ ecbx1000ºLemma‘Õ1ŽŽ‘3„4½L›ÿ}/et‘WXµ–Ǹ6²=“¸hŽ‘ª§²1¸iŽŽ‘å½b˜e–WXan“automorphism“of“a“symmetric“²(µv[Ù;–ª¨kP—;“²)–WX½design“and“let“µF‘ºç½b˜e“a“numb˜er“ofŽŽŸ’Ø–5¹1ŽŽŒ‹ƒ øÒ ýHÐ! ™/ß ý‰Ð!½xe–ÿ}/d›“°p“oints˜of˜the˜automorphism˜µ½.‘™PThen˜e“qualities˜µF–*§¸‘ǵv‘”¹¸‘8à²2µn˜½and˜µF“¸Ÿü ‘ ²š´Ž‘úKŸ£&‰fe.ØŸ*¸k+Bó O!â…cmsy7·Ÿúé¿pŽ‘‘ÉŸú鿉W ñ~ŸA´nŽŽŽŽŽŽŽ‘ð½hold.Ž©óºLemma‘Õ2ŽŽ‘3„4½L›ÿ}/et‘ûµ–Ǹ6²=“¸hŽ‘ª§²1¸iŽŽ‘¥2½b˜e–ûan“involution“of“a“symmetric“²(µv[Ù;–ª¨kP—;“²)–û½design“and“let“µF‘zнb˜e“a“numb˜er“of“xe˜dޤ™œp‘ÿ}/oints–“°of“the“automorphism“µ½.‘™PThenŽŸ!dÌ’™_̵F‘*§¸Ÿèò‘Ç«8ŽŸ ‘Ç>ޤ‘Ç<ŽŸ ‘Ç>Ž¡‘Ç:ŽŽŸ÷‘ª§²1–8à+Ÿü ‘•h´kŽ‘lŸ£&‰fe¾:Ÿ¿˜ŽŽŽŽ‘ ]€µ;‘ãˆf‘or‘ýkP—;‘ª¨“ev[Ùen;ŽŽŸ™›‘‰²1‘8à+Ÿü ‘l´k+B·±1Ž‘lŸ£&‰fe¨Ÿ¿˜‘ôå´ŽŽŽŽ‘GJµ;‘ÿèel2`se:ŽŽŽŽŽŸ!ò‘¹W‘ÿ*¸e–Kâalso“knoš¸èw“that“n˜um˜bšGer“of“p˜oin¸èt“orbits“of“an“automorphism“acting“on“symmetric“(71,15,3)Ž¡design–ö¼m¸èust“bšGe“o˜dd,‘ b˜ecause“the“order“of“(71,15,3)“is“not“a“square“of“an“inš¸èteger,‘ and“that“the“n˜um˜bGerŽ¡of–‰>xed“pGoinš¸èts“of“an“automorphism“of“prime“order“µp“¹acting“on“some“design“is“congruen˜t“²(Ž‘l͵modul2`o‘8àp²)ŽŽŽ¡¹to›¾[n•¸èum“b•Ger˜of˜p“oin•¸èts˜of˜that˜design.‘­2A“ction˜of˜µG˜¹is˜semistandard,‘تso˜to˜determine˜all˜pGossible˜orbitŽ¡distributions–U it“is“sucienš¸èt“to“determine“pGoin˜t“orbit“distributions“²(Ž‘8¯µ!Ÿÿ±1Ž‘|sµ;–ª¨:::;“!Ÿÿ´tŽ‘…V²)ŽŽ‘0ÉÞ¹.ަºLemma‘Õ3ŽŽ‘3„4½L–ÿ}/et›µâµ‘»½b“e˜an˜automorphism˜of˜a˜symmetric˜²(71µ;–ª¨²15µ;“²3)˜½design.‘ÿçIf˜¸jµ[Ù¸j–²=“3½,‘¾othen˜numb‘ÿ}/er˜µFŸÿ±3ŽŽ¡½of–“°xe›ÿ}/d“p˜oints“of“the“automorphism“µ‘satisfy“c˜ondition“µFŸÿ±3Ž‘C‹¸2‘ÇfŽ‘Ç²2µ;–ª¨²5µ;“²8µ;“²11¸gŽŽ‘3n½.ަºProQÂof‘qĹF‘ÿ*¸rom–[ lemma“1“and“condition“µFŸÿ±3Ž›C‹¸‘Dz71‘ª¨(Ž‘Ž7µmod‘8à²3)ŽŽ‘%Ú½¹follo¸èws“µFŸÿ±3Ž˜¸2‘Çf²2µ;–ª¨²5µ;“²8µ;“²11µ;“²14µ;“²17¸g¹.‘$AutomorphismŽ¡of–*symmetric“design“xes“the“same“n•¸èum“bšGer–*of“p˜oin¸èts“and“blo˜c¸èks.‘ð(One“can“not“construct“requiredŽ¡n•¸èum“bšGer–U of“xed“blo˜c¸èks“for“µFŸÿ±3Ž‘C‹¸2‘Çf²14µ;‘ª¨²17¸g¹.¸ަºLemma‘Õ4ŽŽ‘3„4½L–ÿ}/et›–Xµ‘¸½b“e˜an˜automorphism˜of˜a˜symmetric˜²(71µ;–ª¨²15µ;“²3)˜½design.‘¡IIf˜¸jŽ‘]uµ‘!ǸjŽŽ‘q_²=‘Ëé2˜½,‘—then˜numb‘ÿ}/er˜µFŸÿ±2ŽŽ¡½of–“°xe›ÿ}/d“p˜oints“of“the“automorphism“µ‘µw½satisfy“c˜ondition“µFŸÿ±2Ž‘C‹¸2‘ÇfŽ‘Ç²7µ;–ª¨²11µ;“²15¸gŽŽ‘.ª©½.ަºProQÂof‘ 4¹F‘ÿ*¸rom–ëylemma“1“and“condition“µFŸÿ±2Ž›>.¸‘Á»²71‘ª¨(Ž‘Ž7µmod‘8à²2)ŽŽ‘'k-¹follo¸èws“µFŸÿ±2Ž˜¸2‘Á»f²1µ;–ª¨²3µ;“²7µ;“²9µ;“²11µ;“²13µ;“²15µ;“²17¸g¹.‘4ŒF‘ÿ*¸romŽ¡lemma–72“w¸èe“obtain“µFŸÿ±2Ž›¦ ¸‘)™²5¹.‘"ÄCases“µFŸÿ±2Ž˜²=–)™5¹,‘žüµFŸÿ±2Ž˜²=“9¹,‘žüµFŸÿ±2Ž˜²=“13–7¹and“µFŸÿ±2Ž˜²=‘)™17“¹can“bšGe“eliminated“b˜ecauseŽ¡the›U n•¸èum“bGer˜of˜orbits˜of˜automorphism˜µ‘vç¹m“ust˜b•Ge˜o“dd.–q€Therefore,˜µFŸÿ±2Ž‘C‹¸2‘ÇfŽ‘Ç²7µ;–ª¨²11µ;“²15¸gŽŽ‘.ª©¹.“¸ަºLemma‘Õ5ŽŽ‘3„4½L›ÿ}/et–A^µ“½b˜e“an“automorphism“of“a“symmetric“²(71µ;–ª¨²15µ;“²3)–A^½design“and“¸jµ¸j–Dz=“6½.‘}ßF‘ÿ;Èurthermor˜e,‘QÕletŽ¡µFŸÿ±3Ž‘¿´²=‘CA11–Ø5½b›ÿ}/e“a“numb˜er“of“xe˜d“p˜oints“of“the“automorphism“µŸü^ÿ±2Ž‘|s½.‘fàThen“numb˜er“µF‘;Ľof“xe˜d“p˜oints“of“theŽ¡automorphism–“°µ“½satisfy“c‘ÿ}/ondition“µF‘*§¸2‘ÇfŽ‘Ç²1µ;‘ª¨²3¸gŽŽ‘8á½.ަºProQÂof‘ 2޹Let–ÓµdŸÿ´iŽ‘p²(Ž‘ S®µi–Dz=“1µ;–ª¨²2µ;“²3µ;“²4µ;“²5µ;“²6)ŽŽ‘XS1¹bšGe“a“n•¸èum“b˜er–Óof“orbits“of“length“µi“¹of“the“automorphism“µ¹.‘Å™F‘ÿ*¸romŽ¡µFŸÿ±3Ž–C‹²=›ÇµdŸÿ±1Ž‘µS²+‘8à2µdŸÿ±2Ž“²=˜µF‘œo²+‘8à2µdŸÿ±2Ž‘Ñ“¹and‘U µFŸÿ±3Ž“²=˜11–U ¹follo¸èws“µF‘*§¸2˜fŽ‘Ç²1µ;–ª¨²3µ;“²5µ;“²7µ;“²9µ;“²11¸gŽŽ‘Eÿú¹.Ž¡‘Fixed–?ËblošGc¸èks“are“made“of“full“p˜oin¸èt“orbits.‘jdTherefore,›Dif“µF‘*§>‘Dz3¹,˜xed“bloGc¸èks“of“the“automorphismŽ¡µ–\¹conš¸ètain“at“most“one“pGoin˜t“orbit“of“length“six.‘†TIf“µF‘6:¸‘Ò«²3¹,‘]Îthen“xed“bloGc˜ks“of“µ“¹con˜tain“at“most“t˜w˜oŽ¡pGoin¸èt–U orbits“of“length“six.ŽŽŸ’Ø–52ŽŽŒ‹µ øÒ ýHÐ! ™/ß ý‰Ð!‘¹If–sðµF‘^²=›úu11“¹and“µFŸÿ±3Ž‘vè²=˜11¹,‘{¤then“µdŸÿ±2Ž‘vè²=˜0µ;“dŸÿ±3Ž‘vè²=˜0“¹and“µdŸÿ±6Ž‘vè²=˜10¹.‘ÍïSo,‘{¤evš¸èery“xed“bloGc˜k“m˜ust“bGe“made“ofޤ™œnine–OÑxed“pGoinš¸èts“and“one“orbit“of“length“six.‘o»That“led“to“existence“of“elev˜en“dieren˜t“pGoin˜t“orbits“ofŽ¡length–U six,“i.e.›q€to“µdŸÿ±6Ž‘C‹²=‘Ç11¹.˜It“follo¸èws“µF‘*§¸6²=‘Ç11¹.Ž¡‘Let›µxŸÿ±1Ž–|sµxŸÿ±2Ž“µxŸÿ±3Ž“µxŸÿ±6Ž‘û„¹b•Ge˜a˜xed˜blo“c•¸èk˜con“taining˜µxŸÿ´iŽ‘Ó]²(Ž‘ ¶ìµi–Dz=“1µ;–ª¨²2µ;“²3µ;“²6)ŽŽ‘D6!¹pGoin“t˜orbits˜of˜length˜µi¹.‘ïTF‘ÿ*¸or˜example,Ž¡xed–U bloGcš¸èk“²9020“¹con˜tains“nine“xed“pGoin˜ts“and“t˜w˜o“full“pGoin˜t“orbits“of“length“three.Ž¡‘If–S}µF‘*§²=›Ç9“¹and“µFŸÿ±3Ž‘C‹²=˜11¹,‘SÑthen“µdŸÿ±2Ž‘C‹²=˜1µ;“dŸÿ±3Ž‘C‹²=˜2“¹and“µdŸÿ±6Ž‘C‹²=˜9¹.‘pôThe“pšGossible“t¸èyp˜es“of“xed“blo˜c¸èks“are“theŽ¡follo¸èwing:‘q€²9020¹,–U ²9001¹,“²7120¹,“²7101¹,“²6011¹,“²4111¹,“²3021“¹and“²1121¹.Ž¡‘If–°›µF‘Ã'²=›_˜7“¹and“µFŸÿ±3Ž‘Ü ²=˜11¹,‘Çzthen“µdŸÿ±2Ž‘Ü ²=˜2µ;“dŸÿ±3Ž‘Ü ²=˜0“¹and“µdŸÿ±6Ž‘Ü ²=˜10¹.‘ƒñThe“pšGossible“t¸èyp˜es“of“xed“blo˜c¸èks“areŽ¡the–U follo¸èwing:‘q€²7101“¹and“²5201¹.Ž¡‘If–S}µF‘*§²=›Ç5“¹and“µFŸÿ±3Ž‘C‹²=˜11¹,‘SÑthen“µdŸÿ±2Ž‘C‹²=˜3µ;“dŸÿ±3Ž‘C‹²=˜2“¹and“µdŸÿ±6Ž‘C‹²=˜9¹.‘pôThe“pšGossible“t¸èyp˜es“of“xed“blo˜c¸èks“are“theŽ¡follo¸èwing:‘q€²5220¹,–U ²5201¹,“²4111¹,“²3320¹,“²3301¹,“²3021¹,“²2211¹,“²1121“¹and“²0311¹.Ž¡‘In•¸èv“estigating›4–in“tersection˜of˜xed˜bloGc“ks˜one˜can˜obtain˜that˜required˜n“um“b•Ger˜of˜xed˜blo“c¸èks˜canŽ¡not–U bGe“constructed“for“µF‘*§¸2‘ÇfŽ‘Ç²5µ;–ª¨²7µ;“²9¸gŽŽ‘$ª§¹.‘q€¸Ž¡‘¹In–U the“similar“w•¸èa“y›U w“e˜pro“v“e˜the˜follo“wingŽ©™œºLemma‘Õ6ŽŽ‘3„4½L›ÿ}/et–'!µ“½b˜e“an“automorphism“of“a“symmetric“²(71µ;–ª¨²15µ;“²3)–'!½design“and“¸jµ¸j–ÒB²=“6½.‘S£F‘ÿ;Èurthermor˜e,Ž¡let–m•µFŸÿ±3Ž‘C‹²=‘Ç8“½b›ÿ}/e“a“numb˜er“of“xe˜d“p˜oints“of“the“automorphism“µŸü^ÿ±2Ž‘|s½.‘ŒœThen“numb˜er“µF‘Ñ$½of“xe˜d“p˜oints“of“theŽ¡automorphism–“°µ“½satisfy“c‘ÿ}/ondition“µF‘*§¸2‘ÇfŽ‘Ç²0µ;‘ª¨²2¸gŽŽ‘8á½.ަºLemma‘Õ7ŽŽ‘3„4½L›ÿ}/et–jϵ“½b˜e“an“automorphism“of“a“symmetric“²(71µ;–ª¨²15µ;“²3)–jϽdesign“and“¸jµ¸j–Dz=“6½.‘‹°Then–jÏnumb˜er“µFŽ¡½of–“°xe›ÿ}/d“p˜oints“of“the“automorphism“µ“½satisfy“c˜ondition“µF‘*§¸‘Dz3½.ަ‘ºProQÂof‘ ͹Let–¶EµdŸÿ´iŽ‘ ‘²(Ž‘ î µi–Dz=“1µ;–ª¨²2µ;“²3µ;“²4µ;“²5µ;“²6)ŽŽ‘Wˆ¹bšGe“a“n•¸èum“b˜er–¶Eof“orbits“of“length“µi¹.‘”ïF‘ÿ*¸rom“lemmas“3,–ÎŽ4,“5–¶Eand“6Ž¡follo¸èws–š/that“only“remaining“orbit“distribution“for“µF‘Ç>›:8²3“¹is“µdŸÿ±1Ž–¶«²=˜5¹,‘«sµdŸÿ±2Ž“²=˜0µ;‘š/dŸÿ±3Ž“²=˜2–š/¹and“µdŸÿ±6Ž‘¶«²=˜10¹.‘@®AllŽ¡xed–Î%blošGc¸èks“in“this“case“should“b˜e“made“of“three“xed“p˜oin•¸èts,‘é$t“w“o‘Î%p˜oin“t–Î%orbits“of“length“three“and“oneŽ¡pšGoin¸èt–4orbit“of“length“six.‘_‡Fixed“blo˜cš¸èk“of“that“t˜ypšGe“can“o˜ccur“only“once“and“that“pro•¸èv“es–4the“lemma.‘_‡¸ަºLemma‘Õ8ŽŽ‘3„4½Symmetric›¯²(71µ;–ª¨²15µ;“²3)˜½design˜admitting˜an˜action˜of˜µG˜½has˜one˜of˜the˜fol‘‚Ðlowing˜orbitŽ¡distributions:ޤ™œ‘ Ó1.ŽŽŽ‘²(1µ;–ª¨²1µ;“²1µ;“²2µ;“²3µ;“²3µ;“²3µ;“²3µ;“²6µ;“²6µ;“²6µ;“²6µ;“²6µ;“²6µ;“²6µ;“²6µ;“²6)½,Ž¡‘ Ó2.ŽŽŽ‘²(1µ;–ª¨²2µ;“²2µ;“²3µ;“²3µ;“²6µ;“²6µ;“²6µ;“²6µ;“²6µ;“²6µ;“²6µ;“²6µ;“²6µ;“²6)½,ŽŽŸ’Ø–5¹3ŽŽŒ‹#f øÒ ýHÐ! ™/ß ý‰Ð!‘ Ó½3.ŽŽŽ‘²(2µ;–ª¨²3µ;“²3µ;“²3µ;“²3µ;“²3µ;“²6µ;“²6µ;“²6µ;“²6µ;“²6µ;“²6µ;“²6µ;“²6µ;“²6)½.Ž©™œ‘ºProQÂof‘ g¬¹F‘ÿ*¸rom–Pñprevious“lemmas“folloš¸èws“that“required“n˜um˜bšGer“of“xed“blo˜cš¸èks“of“suc˜h“a“design“canޤ™œbGe– constructed“for“the“follo¸èwing“orbit“distributions:‘@(1,–²¸1,“1,“2,“2,“2,“2,“3,“3,“3,“3,“6,“6,“6,“6,“6,“6,“6,Ž¡6),–ˆ(1,“1,“1,“2,“3,“3,“3,“3,“6,“6,“6,“6,“6,“6,“6,“6,“6),“(1,“1,“2,“2,“2,“3,“3,“3,“6,“6,“6,“6,“6,“6,“6,“6,“6),“(1,“1,Ž¡3,–o§3,“3,“6,“6,“6,“6,“6,“6,“6,“6,“6,“6),“(1,“2,“2,“2,“2,“2,“3,“3,“6,“6,“6,“6,“6,“6,“6,“6,“6),“(1,“2,“2,“3,“3,“6,“6,“6,Ž¡6,–o§6,“6,“6,“6,“6,“6),“(2,“2,“2,“2,“3,“3,“3,“3,“3,“6,“6,“6,“6,“6,“6,“6,“6),“(2,“3,“3,“3,“3,“3,“6,“6,“6,“6,“6,“6,“6,“6,Ž¡6).‘²ãF‘ÿ*¸or–jìthat“remaining“cases“w¸èe“should“construct“orbital“structures“satisfying“equalities“(1)“and“(2).Ž¡Because–Hàof“the“large“n•¸èum“bšGer–Hàof“p˜ossibilities,‘KSit“wš¸èas“necessary“to“in˜v˜olv˜e“a“computer“in“this“nal“stepŽ¡of–¢*the“construction“of“orbital“structures.‘XžWith“the“help“of“the“computer“program“bš¸èy“V.“€‚˜epuli€¢“w˜eŽ¡obtained–¦2that“complete“orbit“structures“can“bGe“constructed“only“in“the“cases“from“the“statemen¸èt“ofŽ¡the‘U lemma.‘q€¸ŽŸ/w@Ç3Ž‘ÁConstruction–G\of“designsŽŸ€¹W‘ÿ*¸e–f‘shall“consider“the“construction“of“designs“admitting“an“action“of“µG“¹whic¸èh“correspGonds“to“theŽ¡second–U tš¸èypGe“from“lemma“8.‘q€F‘ÿ*¸or“that“case“w˜e“are“able“to“pro˜v˜e“the“follo˜wingަºTheorem‘Õ1ŽŽ‘;‘ì½Ther–ÿ}/e›"ar“e˜²72˜½mutual‘‚Ðly˜nonisomorphic˜symmetric˜²(71µ;–ª¨²15µ;“²3)˜½designs˜admitting˜an˜actionŽ¡of–[Zthe“cyclic“gr›ÿ}/oup“of“or˜der“6“acting“with“one“xe˜d“p˜oint.‘†‰Betwe˜en“them,‘fžther˜e“ar˜e“²6“½self-dual“designsŽ¡and– ²33“½p‘ÿ}/airs“of“dual›‚Ðly“isomorphic“designs.‘pKExactly“²37“½designs“have“ful˜l“automorphism“gr›ÿ}/oups“of“or˜derŽ¡²24½;‘/»²16–ýÀ½of“these“gr›ÿ}/oups“ar˜e“isomorphic“to“the“gr˜oup“µSŸÿ±3Ž‘oM¸‘òÚµEŸÿ±4Ž‘z3½and“²21“½of“them“ar˜e“isomorphic“to“the“gr˜oupŽ¡µAŸÿ±4Ž‘qq¸‘ôþµZŸÿ±2Ž‘|s½.‘ŽèF‘ÿ;Èurtermor›ÿ}/e,‘z¶²26–tx½designs“have“ful‘‚Ðl“automorphism“gr˜oups“of“or˜der“²48“½isomorphic“to“the“gr˜oupŽ¡µEŸÿ±4Ž‘?o¸‘ÂüµAŸÿ±4Ž‘|s½,‘hP²3–]x½designs“have“ful‘‚Ðl“automorphism“gr›ÿ}/oups“of“or˜der“²168“½isomorphic“to“the“gr˜oup“µEŸÿ±8Ž‘C‹²:‘ǵFŸÿ±21Ž‘ Ö^½andŽ¡²6–“°½designs“have“ful‘‚Ðl“automorphism“gr›ÿ}/oups“of“or˜der“²336“½isomorphic“to“the“gr˜oup“²(Ž‘w?µEŸÿ±8Ž‘C‹²:‘ǵFŸÿ±21Ž‘xæ²)ŽŽ‘0®T¸‘8àµZŸÿ±2Ž‘|s½.ަ‘ºProQÂof‘ €Õ¹First,‘¯Yw•¸èe›jpresen“t˜represen“tativ“es˜of˜pGossible˜t“yp•Ges˜of˜blo“c¸èk˜orbits˜related˜to˜the˜orbitŽ¡distribution›P·(1,–Q™2,“2,“3,“3,“6,“6,“6,“6,“6,“6,“6,“6,“6,“6).‘pIn˜rst˜column˜is˜n•¸èum“bGer˜of˜t“ypGe.‘pIt˜is˜follo“w“edŽ¡with–U length“of“correspšGonding“blo˜cš¸èk“orbit“and“represen˜tativ˜e“of“that“t˜ypGe.Ž¡ŽŸ’Ø–54ŽŽŒ‹0РøÒ ýHÐ! ™/ß ÿOií þ:f4‘¸¹1‘˜1‘ à1–p2“0“3“3“6“0“0“0“0“0“0“0“0“0ޤ™œ‘¸2‘˜1‘ à1–p2“0“0“0“6“6“0“0“0“0“0“0“0“0Ž¡‘¸3‘˜1‘ à0–p0“0“3“0“6“6“0“0“0“0“0“0“0“0Ž¡‘¸4‘˜2‘ à1–p2“0“0“0“3“3“3“3“0“0“0“0“0“0Ž¡‘¸5‘˜2‘ à0–p0“0“3“0“3“3“3“3“0“0“0“0“0“0Ž¡‘¸6‘˜3‘ à1–p2“0“1“1“2“2“2“2“2“0“0“0“0“0Ž¡‘¸7‘˜3‘ à1–p2“0“0“0“2“2“2“2“2“2“0“0“0“0Ž¡‘¸8‘˜3‘ à1–p0“0“2“2“2“2“2“2“2“0“0“0“0“0Ž¡‘¸9‘˜3‘ à1–p0“0“1“1“4“2“2“2“2“0“0“0“0“0Ž¡‘10– „3“1–Â0“0“0“0“4“2“2“2“2“2“0“0“0“0Ž¡‘11– „3“0–Â2“0“2“1“2“2“2“2“2“0“0“0“0“0Ž¡‘12– „3“0–Â0“0“3“0“2“2“2“2“2“2“0“0“0“0Ž¡‘13– „3“0–Â0“0“2“1“4“2“2“2“2“0“0“0“0“0Ž¡‘14– „6“1–Â1“1“2“0“1“1“1“1“1“1“1“1“1“1Ž¡‘15– „6“1–Â1“1“1“1“2“2“1“1“1“1“1“1“0“0Ž¡‘16– „6“1–Â1“1“0“0“3“1“1“1“1“1“1“1“1“1Ž¡‘17– „6“1–Â1“1“0“0“2“2“2“1“1“1“1“1“1“0Ž¡‘18– „6“1–Â1“0“2“0“2“1“1“1“1“1“1“1“1“1Ž¡‘19– „6“1–Â1“0“1“1“3“1“1“1“1“1“1“1“1“0Ž¡‘20– „6“1–Â1“0“1“1“2“2“2“1“1“1“1“1“0“0Ž¡‘21– „6“1–Â1“0“0“0“3“2“1“1“1“1“1“1“1“1Ž¡‘22– „6“1–Â1“0“0“0“2“2“2“2“1“1“1“1“1“0ŽŽ þ:f4’ï23– „6“1–Â0“0“2“0“2“2“1“1“1“1“1“1“1“1ޤ™œ’ï24– „6“1–Â0“0“1“1“3“2“1“1“1“1“1“1“1“0Ž¡’ï25– „6“1–Â0“0“1“1“2“2“2“2“1“1“1“1“0“0Ž¡’ï26– „6“1–Â0“0“0“0“3“2“2“1“1“1“1“1“1“1Ž¡’ï27– „6“1–Â0“0“0“0“2“2“2“2“2“1“1“1“1“0Ž¡’ï28– „6“0–Â2“1“1“0“2“1“1“1“1“1“1“1“1“1Ž¡’ï29– „6“0–Â2“0“1“0“2“2“1“1“1“1“1“1“1“1Ž¡’ï30– „6“0–Â1“1“2“1“2“2“1“1“1“1“1“1“0“0Ž¡’ï31– „6“0–Â1“1“1“0“3“2“2“1“1“1“1“1“0“0Ž¡’ï32– „6“0–Â1“1“1“0“2“2“2“2“2“1“1“0“0“0Ž¡’ï33– „6“0–Â1“0“2“1“3“1“1“1“1“1“1“1“1“0Ž¡’ï34– „6“0–Â1“0“2“1“2“2“2“1“1“1“1“1“0“0Ž¡’ï35– „6“0–Â1“0“1“0“4“1“1“1“1“1“1“1“1“1Ž¡’ï36– „6“0–Â1“0“1“0“3“3“1“1“1“1“1“1“1“0Ž¡’ï37– „6“0–Â1“0“1“0“3“2“2“2“1“1“1“1“0“0Ž¡’ï38– „6“0–Â1“0“1“0“2“2“2“2“2“2“1“0“0“0Ž¡’ï39– „6“0–Â0“0“2“1“3“2“1“1“1“1“1“1“1“0Ž¡’ï40– „6“0–Â0“0“2“1“2“2“2“2“1“1“1“1“0“0Ž¡’ï41– „6“0–Â0“0“1“0“4“2“1“1“1“1“1“1“1“1Ž¡’ï42– „6“0–Â0“0“1“0“3“3“2“1“1“1“1“1“1“0Ž¡’ï43– „6“0–Â0“0“1“0“3“2“2“2“2“1“1“1“0“0Ž¡’ï44– „6“0–Â0“0“1“0“2“2“2“2“2“2“2“0“0“0ŽŽŽŸ#‹D‘F‘ÿ*¸rom–¼-givš¸èen“t˜ypGes“w˜e“shall“construct“orbit“structures“satisfying“equations“(1)“and“(2).‘¦§With“theޤ™œhelp– ßof“the“computer“program“bš¸èy“V.“€‚˜epuli€¢“w˜e“obtained“that“there“are“1464“orbit“structures“for“theŽ¡group–U µGŸýTÍ‘Ç¸ŽŽŸ+3‘Dz=ŽŽŽŽ‘ UNŸ÷æb« Ž‘2µ–Ǹj“µŸü^ÿ±6Ž‘C‹²=“1Ÿ÷æb« ŽŽ‘Cšc¹related“to“the“orbit“distribution“(1,“2,“2,“3,“3,“6,“6,“6,“6,“6,“6,“6,“6,“6,“6).Ž¡‘Let–ûÜus“presenš¸èt“just“one“of“constructed“orbital“structures.‘ e³It“is“ninet˜y-third“obtained“orbitalŽ¡structure–9öand“rst“orbital“structure“that“will“lead“to“(71,15,3)“designs.‘hr(Ordering“of“obtained“orbitalŽ¡structures–”is“describšGed“in“[2].)‘ÝThis“orbit“structure“is“obtained“from“t¸èyp˜e“represen•¸ètativ“es–”as“follo¸èws:Ž¡1,–U 4,“5,“6,“9,“27,“31,“31,“37,“37,“37,“37,“40,“43,“43.ŽŽŸ’Ø–55ŽŽŒ‹Õä6Ž’OÕ”6Ž’`ÕD6ŽŽ‘JW%ŸáL‰ff!}ÏŸµ‘Áh1‘Ž5Ÿzæ„™›ffŽ‘(‚1Ž‘9‚/2Ž‘Jß0Ž‘[3Ž‘l?3Ž‘}€ï6Ž’Ž€Ÿ0Ž’Ÿ€O0Ž’°ÿ0Ž’Á¯0Ž’Ò_0Ž’ã0Ž’ô~¿0Ž’~o0Ž’~0ŽŽ¤™›‘Áh2‘Ž5Ÿzæ„™›ffŽ‘(‚1Ž‘9‚/2Ž‘Jß0Ž‘[0Ž‘l?0Ž‘}€ï0Ž’Ž€Ÿ3Ž’Ÿ€O3Ž’°ÿ3Ž’Á¯3Ž’Ò_0Ž’ã0Ž’ô~¿0Ž’~o0Ž’~0ŽŽ¡‘Áh2‘Ž5Ÿzæ„™›ffŽ‘(‚0Ž‘9‚/0Ž‘Jß0Ž‘[3Ž‘l?0Ž‘}€ï0Ž’Ž€Ÿ3Ž’Ÿ€O3Ž’°ÿ0Ž’Á¯0Ž’Ò_3Ž’ã3Ž’ô~¿0Ž’~o0Ž’~0ŽŽ¡‘Áh3‘Ž5Ÿzæ„™›ffŽ‘(‚1Ž‘9‚/0Ž‘Jß2Ž‘[1Ž‘l?1Ž‘}€ï0Ž’Ž€Ÿ2Ž’Ÿ€O2Ž’°ÿ0Ž’Á¯0Ž’Ò_0Ž’ã0Ž’ô~¿2Ž’~o2Ž’~2ŽŽ¡‘Áh3‘Ž5Ÿzæ„™›ffŽ‘(‚1Ž‘9‚/0Ž‘Jß0Ž‘[1Ž‘l?1Ž‘}€ï0Ž’Ž€Ÿ0Ž’Ÿ€O0Ž’°ÿ2Ž’Á¯2Ž’Ò_2Ž’ã2Ž’ô~¿4Ž’~o0Ž’~0ŽŽ¡‘Áh6‘Ž5Ÿzæ„™›ffŽ‘(‚1Ž‘9‚/0Ž‘Jß0Ž‘[0Ž‘l?0Ž‘}€ï2Ž’Ž€Ÿ1Ž’Ÿ€O1Ž’°ÿ1Ž’Á¯1Ž’Ò_2Ž’ã2Ž’ô~¿0Ž’~o2Ž’~2ŽŽ¡‘Áh6‘Ž5Ÿzæ„™›ffŽ‘(‚0Ž‘9‚/1Ž‘Jß1Ž‘[0Ž‘l?1Ž‘}€ï1Ž’Ž€Ÿ2Ž’Ÿ€O0Ž’°ÿ2Ž’Á¯0Ž’Ò_3Ž’ã1Ž’ô~¿1Ž’~o1Ž’~1ŽŽ¡‘Áh6‘Ž5Ÿzæ„™›ffŽ‘(‚0Ž‘9‚/1Ž‘Jß1Ž‘[0Ž‘l?1Ž‘}€ï1Ž’Ž€Ÿ0Ž’Ÿ€O2Ž’°ÿ0Ž’Á¯2Ž’Ò_1Ž’ã3Ž’ô~¿1Ž’~o1Ž’~1ŽŽ¡‘Áh6‘Ž5Ÿzæ„™›ffŽ‘(‚0Ž‘9‚/1Ž‘Jß0Ž‘[1Ž‘l?0Ž‘}€ï1Ž’Ž€Ÿ2Ž’Ÿ€O0Ž’°ÿ0Ž’Á¯2Ž’Ò_1Ž’ã1Ž’ô~¿2Ž’~o3Ž’~1ŽŽ¡‘Áh6‘Ž5Ÿzæ„™›ffŽ‘(‚0Ž‘9‚/1Ž‘Jß0Ž‘[1Ž‘l?0Ž‘}€ï1Ž’Ž€Ÿ0Ž’Ÿ€O2Ž’°ÿ2Ž’Á¯0Ž’Ò_1Ž’ã1Ž’ô~¿2Ž’~o1Ž’~3ŽŽ¡‘Áh6‘Ž5Ÿzæ„™›ffŽ‘(‚0Ž‘9‚/0Ž‘Jß1Ž‘[1Ž‘l?0Ž‘}€ï2Ž’Ž€Ÿ1Ž’Ÿ€O1Ž’°ÿ3Ž’Á¯1Ž’Ò_0Ž’ã2Ž’ô~¿1Ž’~o2Ž’~0ŽŽ¡‘Áh6‘Ž5Ÿzæ„™›ffŽ‘(‚0Ž‘9‚/0Ž‘Jß1Ž‘[1Ž‘l?0Ž‘}€ï2Ž’Ž€Ÿ1Ž’Ÿ€O1Ž’°ÿ1Ž’Á¯3Ž’Ò_2Ž’ã0Ž’ô~¿1Ž’~o0Ž’~2ŽŽ¡‘Áh6‘Ž5Ÿzæ„™›ffŽ‘(‚0Ž‘9‚/0Ž‘Jß0Ž‘[1Ž‘l?2Ž‘}€ï0Ž’Ž€Ÿ1Ž’Ÿ€O1Ž’°ÿ2Ž’Á¯2Ž’Ò_1Ž’ã1Ž’ô~¿0Ž’~o2Ž’~2ŽŽ¡‘Áh6‘Ž5Ÿzæ„™›ffŽ‘(‚0Ž‘9‚/0Ž‘Jß0Ž‘[0Ž‘l?1Ž‘}€ï2Ž’Ž€Ÿ3Ž’Ÿ€O1Ž’°ÿ1Ž’Á¯1Ž’Ò_0Ž’ã2Ž’ô~¿2Ž’~o0Ž’~2ŽŽ¡‘Áh6‘Ž5Ÿzæ„™›ffŽ‘(‚0Ž‘9‚/0Ž‘Jß0Ž‘[0Ž‘l?1Ž‘}€ï2Ž’Ž€Ÿ1Ž’Ÿ€O3Ž’°ÿ1Ž’Á¯1Ž’Ò_2Ž’ã0Ž’ô~¿2Ž’~o2Ž’~0ŽŽŽŽŽ ¾c+‘F‘ÿ*¸or–‘Ònal“step“of“the“construction,›àÿcalled“indexing,˜wš¸èe“dev˜elopGed“computer“programs.‘'—Duringޤ™œconstruction–¤_of“symmetric“designs“wš¸èe“shall“use“elemen˜ts“of“a“normalizer“of“an“automorphism“groupŽ¡in–Sthe“group“µS‘Z¥²=‘ǵS›“²(¸P‘Ò}²)–=¸“µS˜²(¸BMÛ²)–S¹to“a•¸èv“oid–Sconstruction“of“m¸èutually“isomorphic“designs“(see“[1],‘â[2]).‘W‘W‘ÿ*¸eŽ¡shall›Ìc•¸èhec“k˜all˜p•Gossibilities˜corresp“onding˜to˜obtained˜orbital˜structures.‘CÕThere˜is˜a˜v•¸èery˜large˜n“um“bGerŽ¡of–î‰pšGossibilities.‘ONF‘ÿ*¸or“example,‘smallest“n•¸èum“b˜er–î‰of“p˜ossibilities“for“some“orbit“of“length“six“of“presen¸ètedŽ¡ninetš¸èy-third–µcstructure“is“²52–ª¨488“000–g‘¸“²16‘ª¨000–y¸“²2Ÿü^ÿ±15Ž‘ .I¹and–µcbiggest“n˜um˜bšGer“of“p˜ossibilities“for“some“orbitŽ¡of–;Vlength“six“is“²1–ª¨594“323“000“000–Ǹ“²48–ª¨650“000–I¸“²2Ÿü^ÿ±15Ž‘xæ¹.‘hçThis–;VprošGcess“of“indexing“will“tak¸èe“to˜o“m•¸èuc“h‘;Vtime,Ž¡evš¸èen–²Æif“w˜e“in˜v˜olv˜e“computers.‘ŠrTherefore,‘Ê/w˜e“proGceed“b˜y“lifting‘ˆobtained“orbital“structures“for“theŽ¡group–;%µGŸýTÍ‘Ç¸ŽŽŸ+3‘Dz=ŽŽŽŽ‘ UNŸ÷æb« Ž‘2µ–Ǹj“µŸü^ÿ±6Ž‘C‹²=“1Ÿ÷æb« ŽŽ‘C€h¹to“orbital“structures“for“the“cyclic“group“Ÿ÷æb« Ž‘ô µŸü^ÿ±3Ž‘|sŸ÷æb« ŽŽ‘¹with“the“assumption“that“theyŽ¡admit–[µŸü^ÿ±2Ž‘ŽÎ¹as“an“automorphism.‘©1As“a“result“w¸èe“got“that“248“orbit“structures“can“bGe“lifted“to“orbitŽ¡structures–\tfor“the“cyclic“group“Ÿ÷æb« Ž‘XµŸü^ÿ±3Ž‘|sŸ÷æb« ŽŽ›Ò²¹with“the“assumption“that“they“admit“Ÿ÷æb« Ž‘XµŸü^ÿ±2Ž‘|sŸ÷æb« ŽŽ˜¹as“an“automorphism.Ž¡Because–¤zof“their“large“n•¸èum“bGer,‘¸Qw“e›¤zpresen“t˜just˜ordinal˜n“um“b•Gers˜of˜that˜structures˜corresp“onding˜toŽ¡the–U ordering“describGed“in“[2].ŽŽŸ’Ø–56ŽŽŒ‹H> øÒ ýHÐ! ™/ß ý‰Ð!¤™œ‘¹3‘Sà117–© 166“232“311“390“481“577“624“680“774“970“1308Ž¡‘4‘Sà119–© 167“233“321“391“482“579“625“682“783“979“1327Ž¡‘31‘þÀ120–© 168“234“323“392“484“581“627“684“799“981“1382Ž¡‘33‘þÀ121–© 171“236“327“393“485“582“632“687“830“983“1404Ž¡‘35‘þÀ122–© 174“240“328“401“486“583“633“691“839“991“1425Ž¡‘61‘þÀ124–© 177“243“330“405“487“584“637“694“856“993“1428Ž¡‘64‘þÀ127–© 180“246“331“407“497“589“639“697“864‘Sà996‘ ÿ`1452Ž¡‘66‘þÀ129–© 194“255“342“421“498“591“643“700“884“1012‘ T€1463Ž¡‘91‘þÀ130–© 196“261“343“439“508“592“646“702“885“1070Ž¡‘93‘þÀ132–© 197“270“346“449“513“593“649“703“888“1076Ž¡‘96‘þÀ141–© 202“290“362“450“524“594“650“708“908“1106Ž¡‘97‘þÀ142–© 205“291“363“453“555“595“654“709“910“1124Ž¡‘100–© 147“206“292“366“456“558“597“655“719“911“1138Ž¡‘102–© 148“207“293“368“457“559“598“656“720“932“1150Ž¡‘104–© 151“208“294“369“461“563“600“661“722“933“1168Ž¡‘105–© 152“211“296“373“463“566“611“664“726“936“1227Ž¡‘107–© 159“217“297“374“464“568“612“665“730“955“1237Ž¡‘110–© 161“226“299“385“467“569“613“667“732“956“1261Ž¡‘111–© 164“229“303“386“468“571“618“671“742“967“1282Ž¡‘114–© 165“231“309“388“474“574“622“678“747“968“1292Ž¡¡‘F‘ÿ*¸or–îÌeacš¸èh“orbit“structure“that“pass‘Äin“this“rst“step“of“indexing“w˜e“obtain“more“than“v˜e“and“lessŽ¡than–€ sixtš¸èy-one“orbit“structures“for“the“group“Ÿ÷æb« Ž‘9„µŸü^ÿ±3Ž‘|sŸ÷æb« ŽŽ‘šj¹.‘ôIn“nal“step“of“indexing“w˜e“deal“with“3270“orbitŽ¡structures–&‡for“the“group“Ÿ÷æb« Ž‘ßkµŸü^ÿ±3Ž‘|sŸ÷æb« ŽŽ‘@Q¹.‘aøIn“this“case,‘/Ùthe“n•¸èum“bšGer–&‡of“p˜ossibilities“for“eacš¸èh“ro˜w“of“orbit“structureŽ¡is–U less“than“²2Ÿü^ÿ±15Ž‘xæ¹.Ž¡‘Checš¸èking––xall“pGossibilities“w˜e“obtained“that“only“30“of“1464“orbit“structures“for“cyclic“group“of“orderŽ¡six–ÞQcan“bšGe“lifted“to“the“design“with“parameters“(71,15,3).‘ The“ordinal“n•¸èum“b˜ers–ÞQof“those“structuresŽ¡are‘¤µ93,–¸›100,“114,“119,“122,“124,“132,“152,“159,“164,“168,“177,“206,“243,“468,“484,“485,“568,“571,“577,Ž¡589,–U 594,“643,“649,“665,“691,“702,“885,“910“and“996.ŽŽŸ’Ø–57ŽŽŒ‹X| øÒ ýHÐ! ™/ß ý‰Ð!‘¹After–8Àeliminating“isomorphic“copies“(using“program“bš¸èy“V.“€‚˜epuli€¢)“w˜e“got“exactly“72“non-isomorphicޤ™œdesigns.‘XnBet•¸èw“een– éthem,‘ôthere“are“six“self-dual“designs“and“thirt¸èy-three“pairs“of“dually“isomorphic“de-Ž¡signs.‘~GF‘ÿ*¸or–Ybexample,‘šsfrom“presenš¸èted“ninet˜y-third“orbital“structure“w˜e“obtain“eigh˜t“non-isomorphicŽ¡designs–Ì1-8“presenš¸èted“bGelo˜w.‘×öDuals“of“those“eigh˜t“designs“are“obtained“from“eigh˜t“h˜undred“andŽ¡eigh•¸èt“y-fth‘U structure.Ž¡‘W‘ÿ*¸e–}épresenš¸èt“six“self-dual“designs“and“one“design“for“eac˜h“obtained“pair“of“dually“isomorphic“designs.Ž¡F–ÿ*¸or‘ä¡brevitš¸èy“,‘represen˜tativ˜es–ä¡of“common“line“orbits“of“length“one,‘t˜w˜o“or“three“for“more“designs“areŽ¡written–(yonly“once.‘bžIn“those“cases,‘1grst“n•¸èum“bGer–(yin“eacš¸èh“ro˜w,›1gnamely“µl2`¹,˜is“the“length“of“line“orbit.‘bžIt“isŽ¡follo•¸èw“ed›eòb“y˜the˜line˜that˜is˜represen“tativ“e˜of˜line˜orbit˜and˜others˜can˜bGe˜obtained˜b“y˜applying˜µl‘vw¸‘D²1Ž¡¹times‘U automorphismŽ¡‘µ–Dz=“(1µ;–ª¨²2)(3µ;“²4)(5µ;“²6µ;“²7)(8µ;“²9µ;“²10)(11µ;“²16µ;“²12µ;“²14µ;“²13µ;“²15)(17µ;“²22µ;“²18µ;“²20µ;“²19µ;“²21)(23µ;“²28µ;“²24µ;“²26µ;“²25µ;“²27)Ž¡‘(29µ;–ª¨²34µ;“²30µ;“²32µ;“²31µ;“²33)(35µ;“²40µ;“²36µ;“²38µ;“²37µ;“²39)(41µ;“²46µ;“²42µ;“²44µ;“²43µ;“²45)(47µ;“²52µ;“²48µ;“²50µ;“²49µ;“²51)Ž¡‘(53µ;–ª¨²58µ;“²54µ;“²56µ;“²55µ;“²57)(59µ;“²64µ;“²60µ;“²62µ;“²61µ;“²63)(65µ;“²70µ;“²66µ;“²68µ;“²67µ;“²69)¹.Ž¡‘Represen•¸ètativ“es–ÚÎfor“bloGcš¸èk“orbits“of“length“six“are“giv˜en“for“eac˜h“design“separately“and“other“bloGc˜ksŽ¡from–U these“orbits“can“bGe“obtain“bš¸èy“applying“v˜e“times“giv˜en“automorphism“µ¹.Ž¡¡‘common–U lines“for“designs“1-39:Ž¡‘1‘þÀ0–ª@1“2“5“6“7“8“9–U 10“11“12“13“14“15“16Ž¡‘2‘þÀ0–ª@1“2–U 17“18“19“23“24“25“29“30“31“35“36“37Ž¡‘2‘þÀ5–ª@6“7–U 17“18“19“26“27“28“41“42“43“47“48“49Ž¡¡‘common–U lines“for“designs“1-24:Ž¡‘3‘þÀ0–ª@3“4“5“8–U 17“20“23“26“53“56“59“62“65“68Ž¡‘3‘þÀ0–ª@5“8–U 29“32“35“38“41“44“47“50“54“55“57“58Ž¡¡‘design‘U 1:ŽŸZ/£Ÿ½Â‘0–U 11“14“17“23“32“38“42“46“49“51“60“61“66“67ޤ©‘1–ª@3“8–U 11“18“22“30“34“41“42“44“52“53“60“70Ž¡‘1–ª@4“8–U 11“25“27“37“39“46“47“48“50“56“61“69Ž¡‘1‘ª@5–U 14“18“22“37“39“45“49“54“55“59“62“64“66ŽŽŸ½Â’ï1‘ª@5–U 14“25“27“30“34“43“51“57“58“63“65“67“68ޤ©’ï3‘ª@5–U 13“15“22“24“29“30“32“40“48“49“56“61“63Ž¡’ï3‘ª@5–U 12“16“18“28“34“35“36“38“45“46“56“67“69Ž¡’ï5‘ª@9–U 10“17“26“30“31“39“40“44“50“60“64“67“69ŽŽŽŽŸ’Ø–58ŽŽŒ‹ a  øÒ ýHÐ! ™/ß ý‰Ð!‘¹8–U 12“13“17“20“22“27“31“36“48“52“55“57“66“67ŽŽ’ï8–U 12“13“18“23“25“26“33“40“42“46“54“58“63“64ŽŽŽ¤#‹D‘design‘U 2:Ž©nXŸ©™‘0–U 11“14“17“23“32“38“42“46“49“51“60“61“66“67ޤ™œ‘1–ª@3“8–U 11“18“22“30“34“41“42“44“52“53“60“70Ž¡‘1–ª@4“8–U 11“25“27“37“39“46“47“48“50“56“61“69Ž¡‘1‘ª@5–U 14“19“21“37“39“42“52“57“58“59“62“64“66Ž¡‘1‘ª@5–U 14“24“28“30“34“46“48“54“55“63“65“67“68ŽŽŸ©™’ï3‘ª@5–U 12“16“22“24“29“32“33“37“48“49“56“60“64ޤ™œ’ï3‘ª@5–U 13“15“18“28“31“35“38“39“45“46“56“66“70Ž¡’ï5‘ª@9–U 10“17“26“33“34“36“37“44“50“61“63“66“70Ž¡’ï8–U 12“13“17“20“22“27“31“36“48“52“55“57“66“67Ž¡’ï8–U 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14“19“21“34“36“44“48“49“51“57“67“68ŽŽŸÑ]‘’ï1‘ª@5–U 11“22“27“30“31“49“50“55“56“60“62“63“67ޤQ8’ï1‘ª@8–U 14“18“25“39“40“42“43“45“50“53“58“61“62Ž¡’ï3‘ª@5–U 15“16“18“20“26“36“37“40“51“58“60“64“67ŽŽŽŽŸ’Ö]16ŽŽŒ‹³™ øÒ ýHÐ! ™/ß ýŸi½Ÿêfd‘¹3‘ª@8–U 12“16“24“25“28“30“32“38“46“48“49“58“63ŽŸ™œ‘5‘ª@9–U 10“19“28“30“32“35“39“45“51“62“64“68“70ŽŽŸêfd’ï5–U 12“13“18“31“32“34“36“38“43“45“52“56“57“66ŽŸ™œ’ï8–U 12“16“17“19“22“23“27“33“45“53“54“64“66“67ŽŽŽ¤!H¥‘design‘U 37:Ž©lyŸ©™‘0–U 11“14“17“26“29“38“45“46“48“52“60“61“66“70ޤ™œ‘1–ª@3“5–U 11“24“30“34“40“42“44“47“55“66“67“70Ž¡‘1–ª@4“8–U 14“22“24“28“30“48“49“52“58“59“63“65Ž¡‘1‘ª@5–U 11“18“22“28“36“45“54“56“61“62“64“68“69Ž¡‘1‘ª@8–U 14“18“34“36“40“41“43“46“50“51“53“57“60ŽŽŸ©™’ï3‘ª@5–U 15“16“17“22“31“36“38“39“49“51“57“63“70ޤ™œ’ï3‘ª@8–U 13“15“20“24“27“28“35“37“43“45“57“61“66Ž¡’ï5‘ª@9–U 10“18“27“31“32“35“40“46“52“62“63“65“66Ž¡’ï5–U 12“13“25“26“29“30“33“40“43“51“56“58“61“63Ž¡’ï8–U 12“16“18“19“21“23“32“34“45“49“56“58“66“70ŽŽŽ¡‘design‘U 38:ަŸ©™‘0–U 11“14“17“26“29“38“45“46“49“51“60“61“66“70ޤ™œ‘1–ª@3“5–U 11“24“30“34“40“47“49“52“55“60“62“69Ž¡‘1–ª@4“8–U 14“22“24“28“30“41“45“48“58“67“68“70Ž¡‘1‘ª@5–U 11“18“22“28“36“44“54“56“61“63“64“65“66Ž¡‘1‘ª@8–U 14“18“34“36“40“42“43“46“50“51“53“57“59ŽŽŸ©™’ï3‘ª@5–U 15“16“25“26“29“36“39“40“43“45“58“63“67ޤ™œ’ï3‘ª@8–U 12“16“18“20“21“28“35“37“49“51“58“61“69Ž¡’ï5‘ª@9–U 10“18“27“31“32“35“40“45“51“62“64“68“70Ž¡’ï5–U 12“13“17“22“30“31“33“38“43“51“56“57“67“69Ž¡’ï8–U 12“16“22“24“26“27“29“31“42“52“53“54“60“64ŽŽŽ¡‘design‘U 39:ަŸ©™‘0–U 11“14“17“23“32“38“42“46“49“51“60“61“66“67ޤ™œ‘1–ª@3“5–U 12“24“37“38“40“43“45“49“55“60“62“69Ž¡‘1–ª@4“8–U 15“23“28“31“33“41“48“51“52“55“60“70Ž¡‘1‘ª@5–U 12“19“21“28“29“51“54“56“61“63“64“65“66Ž¡‘1‘ª@8–U 15“19“20“34“36“42“45“46“47“56“57“67“69ŽŽŸ©™’ï3‘ª@5–U 14“16“22“25“27“29“33“36“42“55“65“67“70ޤ™œ’ï3‘ª@8–U 13“14“17“18“21“28“36“40“50“52“58“61“69Ž¡’ï5‘ª@9–U 10“17“23“33“34“37“39“41“50“63“64“67“69Ž¡’ï5–U 11“13“21“31“33“34“35“36“43“49“51“53“57“59Ž¡’ï8–U 11“16“22“23“24“27“30“40“41“43“56“57“61“63ŽŽŽ¡‘Designs–U 25,“26,“29,“36,“37“and“38“are“self-dual.ޤ™œ‘Program–tÙbš¸èy“V.€‚˜epuli€¢“ga˜v˜e“us“generators“of“full“automorphism“groups“for“constructed“designs.‘&¾W‘ÿ*¸eŽ¡determine–Ç™the“full“automorphism“groups“using“GAP‘Çu(see“[7]).‘BSF‘ÿ*¸ull“automorphism“groups“of“designs“1,Ž¡2,–±m3,“4,“6,“7,“8,“10,“11,“12,“14,“15–ˆand“16“are“isomorphic“to“group“µEŸÿ±4Ž‘ý¸‘ŸŠµAŸÿ±4Ž‘ô¹and“full“automorphism“groupsŽ¡of–sádesigns“5,› î9“and“13“are“isomorphic“to“group“²(Ž‘WpµEŸÿ±8Ž‘C‹²:‘ǵFŸÿ±21Ž‘xæ²)ŽŽ‘-Ëí¸vHµZŸÿ±2Ž‘|s¹.‘&kF‘ÿ*¸urthermore,˜full“automorphism“groupsŽ¡of–ƒŸdesigns“17,–>18,“19,“20,“21,“22,“23–ƒŸand“24“are“isomorphic“to“group“µSŸÿ±3Ž‘ÔT¸‘WáµEŸÿ±4Ž‘¹and“full“automorphismŽ¡groups–ê-of“designs“25,–ÿ26,“27,“28,“29,“30,“31,“32,“33,“35,“37,“38–ê-and“39“are“isomorphic“to“group“µAŸÿ±4Ž‘ß`¸‘bíµZŸÿ±2Ž‘|s¹.Ž¡Finally‘ÿ*¸,–U full“automorphism“groups“of“designs“34“and“36“are“isomorphic“to“group“µEŸÿ±8Ž‘C‹²:‘ǵFŸÿ±21Ž‘xæ¹.‘q€¸ŽŽŸ’Ö]¹17ŽŽŒ‹¾y øÒ ýHÐ! ™/ß ý‰Ð!‘ºRemark–ȹT‘ÿ*¸riplanes“from“theorem“1“ha¸èving“full“automorphism“groups“of“orders“168“and“336“areޤ™œisomorphic–Q‰to“symmetric“(71,15,3)“designs“describGed“in“[3].‘f»Constructed“triplanes“also“include“allŽ¡(71,15,3)–`designs“describGed“in“[4].‘?Precisely‘ÿ*¸,‘<°design“3“wš¸èas“also“kno˜wn.‘?So,‘<°triplanes“from“theoremŽ¡1–y,include“all“11“knoš¸èwn“triplanes“with“parameters“(71,15,3).‘Ý£W‘ÿ*¸e“constructed“sixt˜y-one“new“m˜utuallyŽ¡non-isomorphic–U symmetric“(71,15,3)“designs.ŽŸ/w@ÇReferencesŽŸ€¹[1]ŽŽ‘D.›‚ßCrnk•¸èo“vi€¢˜and˜S.˜Ruk‘ÿqÐa“vina,‘ŽOSymmetric˜(66,26,10)˜designs˜ha“ving˜µF‘crGobŸÿ±55Ž‘ ûŹas˜an˜automorphismŽ¡‘group,–U Glasnik“matemati€£¸èki,“to“appGearŽ©™œ[2]ŽŽ‘V.–þ=€‚š¸èepuli€¢,‘žOn“symmetric“bloGc˜k“designs“(40,13,4)“with“automorphisms“of“order“5,‘žDiscrete“Math.Ž¡‘128–U (1994)“no.“13,“4560.ަ[3]ŽŽ‘M.–"¨Garapi€¢,›V Construction“of“some“new“triplanes.“Ph.“D.“Thesis,˜Univ•¸èersit“y–"¨of“Zagreb,˜Zagreb,Ž¡‘Croatia,‘U 1993.ަ[4]ŽŽ‘W.H.–ˆ0Haemers,‘ÔôEigenš¸èv‘ÿqÐalue“T‘ÿ*¸ec˜hniques“in“Design“and“Graph“Theory‘ÿ*¸,‘ÔôMathematisc˜h“Cen˜trum,Ž¡‘Amsterdam,‘U 1980.ަ[5]ŽŽ‘E.–U Lander,“Symmetric“Designs:‘q€An“Algebraic“Approacš¸èh,“Cam˜bridge“Univ˜ersit˜y“Press“(1983).ަ[6]ŽŽ‘R.–°åA.“Mathon“and“A.“Rosa,›Ö²2– Ƹ“²(Ž‘Uµv[Ù;–ª¨kP—;“²)ŽŽ‘) ½¹Designs–°åof“Small“Order,˜in“The“CR¸èC‘°‹Handb•Go“ok‘°åofŽ¡‘Comš¸èbinatorial–U Designs,“(ed.“C.“J.“ColbGourn“and“J.“H.“Dinitz),“CR˜C“Press“(1996),“3-41.ަ[7]ŽŽ‘M.–;VSc¸èhoGenert“et“al.,›@~GAP‘;O-“Groups,˜Algorithms“and“programming,˜Lehrstuhl“D‘;Ofur“Mathematik,Ž¡‘Rheinisc•¸èh›U W‘ÿ*¸estfallisc“he˜T‘ÿ*¸ec“hnishe˜HoGc“hsh“ule,˜Aac“hen˜1995.ŽŽŸ’Ö]18ŽŽŒøÊƒ’À;èøÒ¶, ó·ág£ff cmmi12ó&Lt$ffffecbx1440ó©±Ê cmsy9ó5ùž" cmmi9ó¹Aa¨cmr6óo´‹Ç cmr9ó‹–uÌ ecbx0900óÙ.œŒ ecrm0900ó½HЃ ecti1000ó¥!¢N ecbx1200ó]fŒ ecbx1000ó 1ê± ecrm1000ó !",š cmsy10ó O!â…cmsy7ó  b> cmmi10ó 0e—rcmmi7óKñ`y 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