Abstract
Research on the existence of large sets of Kirkman triple systems (LKTS) extends from the mid-eighteen hundreds to the present. In this paper we review known direct approaches of constructing LKTS and present new ideas of direct constructions. We finally prove the existence of an LKTS(v) where \(v \in \{69,141,165,213,285,309,333\}\). Combining recursive constructions yields several new infinite classes.
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Chang Y., Ge G.: Some new large sets of KTS\((v)\). Ars Comb. 51, 306–312 (1999).
Chang Y., Zhou J.: Large sets of Kirkman triple systems and related designs. J. Comb. Theory (A) 120, 649–670 (2013).
Denniston R.H.F.: Sylvester’s problem of the 15 schoolgirls. Discret. Math. 9, 229–233 (1974).
Denniston R.H.F.: Further cases of double resolvability. J. Comb. Theory (A) 26, 298–303 (1979).
Denniston R.H.F.: Four double resolvable complete three-designs. Ars Comb. 7, 265–272 (1979).
Ge G.: More large sets of KTS\((v)\). J. Comb. Math. Comb. Comput. 49, 211–214 (2004).
Kirkman T.P.: Note on an unanswered prize question. Camb. Dublin Math. J. 5, 255–262 (1850).
Lu J.: On large sets of disjoint Steiner triple systems, I–III. J. Comb. Theory (A) 34, 140–182 (1983).
Lu J.: On large sets of disjoint Steiner triple systems, IV–V. J. Comb. Theory (A) 37, 136–192 (1984).
Ray-Chaudhuri D.K., Wilson R.M.: Solution of Kirkman’s schoolgirl problem. Proc. Symp. Pure Math. 19, 187–204 (1971).
Schreiber S.: Covering all triples on \(n\) marks by disjoint Steiner systems. J. Comb. Theory (A) 15, 347–350 (1973).
Schreiber S., quoted by Hanani H.: Resolvable designs. In: Colloquio Internazionale Sulle Teorie Combinatorie (Atti dei Convegni Lincei), Rome, Tomo II, vol. 17, pp. 249–252 (1976).
Teirlinck L.: A completion of Lu’s determination of the spectrum for large sets of disjoint Steiner triple systems. J. Comb. Theory (A) 57, 302–305 (1991).
Wilson R.M.: Some partitions of all triples into Steiner triple systems. In: Lecture Notes in Mathematics, vol. 411, pp. 267–277 (1974).
Wang C., Shi C.: Large sets of Kirkman triple systems with order \(q^n+2\). arXiv:1307.3019 (2013).
Zhang S., Zhu L.: Transitive resolvable idempotent symmetric quasigroups and large sets of Kirkman triple systems. Discret. Math. 247, 215–223 (2002).
Zhou J., Chang Y.: New results on large sets of Kirkman triple systems. Des. Codes Cryptogr. 55, 1–7 (2010).
Acknowledgments
Supported by NSFC Grants 11271042, 11431003 and 11571034.
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Communicated by L. Teirlinck.
Appendix
Appendix
We list all the base blocks of \((q-1,3,\{2,3\},2)\) disjoint difference family over \(\mathbb {Z}_{q-1}\) relative to \(H=\{0,{q-1\over 3},{2(q-1)\over 3}\}\) in Lemma 3.4. Note that in each base block set the three pairs \(\{x_i,y_i\}\) \((i=1,2,3)\) are listed in order; they give rise to three blocks \(\{\infty _1,g^{x_1},g^{y_1}\},\) \(\{\infty _2,g^{x_2},g^{y_2}\}\) and \(\{0,g^{x_3},g^{y_3}\}\) as in Theorem 3.2.
\((1)\ q=67, g=2\) | ||||
{13,42} | {20,33} | {39,38} | ||
{46,36,6} | {1,24,32} | {0,2,54} | {3,16,62} | {40,12,8} |
{19,10,58} | {41,60,18} | {5,52,7} | {17,28,22} | {15,64,48} |
{51,43,57} | {25,14,63} | {31,49,34} | {59,30,35} | {9,29,50} |
{23,53,27} | {65,11,26} | {4,61,45} | {55,21,56} | {44,47,37} |
\((2)\ q=139, g= 2\) | ||||
{92,45} | {27,62} | {136,137} | ||
{1,116,20} | {74,134,72} | {41,102,120} | {0,64,90} | {3,6,104} |
{2,30,124} | {86,34,28} | {76,100,49} | {4,129,130} | {42,52,56} |
{50,88,80} | {82,32,10} | {87,128,58} | {61,40,8} | {51,60,114} |
{44,78,63} | {77,66,46} | {94,12,133} | {84,99,48} | {107,118,13} |
{127,59,18} | {5,101,96} | {126,55,119} | {122,79,39} | {108,33,71} |
{57,9,112} | {105,24,69} | {123,113,110} | {135,31,15} | {37,25,70} |
{117,35,11} | {21,109,81} | {98,93,17} | {95,121,22} | {115,106,29} |
{16,23,131} | {91,26,111} | {53,36,125} | {43,47,68} | {83,85,14} |
{97,54,103} | {65,38,7} | {132,19,73} | {89,75,67} |
\((3)\ q=163, g= 2\) | ||||
{144,81} | {139,156} | {113,112} | ||
{101,152,64} | {15,106,132} | {1,62,54} | {36,98,122} | {73,4,46} |
{16,72,23} | {0,28,22} | {3,130,90} | {103,74,84} | {51,18,76} |
{116,31,44} | {47,88,42} | {17,94,60} | {125,146,108} | {2,52,134} |
{30,69,66} | {12,25,80} | {48,135,68} | {57,8,26} | {75,34,32} |
{41,148,70} | {102,118,114} | {141,10,24} | {40,104,136} | {58,160,21} |
{117,14,124} | {126,147,82} | {142,50,133} | {6,120,129} | {100,151,65} |
{110,39,121} | {158,53,93} | {38,19,11} | {45,7,154} | {89,92,91} |
{49,105,137} | {55,77,67} | {5,161,97} | {61,150,35} | {95,145,109} |
{9,56,119} | {87,111,83} | {43,143,85} | {149,131,71} | {33,79,99} |
{13,29,86} | {27,155,96} | {153,123,128} | {107,59,140} | {127,37,138} |
{115,159,20} | {78,157,63} |
\((4)\ q=211, g = 2\) | ||||
{47,68} | {178,19} | {0,43} | ||
{1,84,131} | {137,59,159} | {129,141,115} | {85,161,181} | {207,83,167} |
{163,21,5} | {81,9,209} | {199,169,77} | {11,75,192} | {201,89,104} |
{55,6,91} | {86,205,187} | {118,101,39} | {133,175,4} | {183,79,148} |
{93,190,149} | {41,13,40} | {97,184,121} | {135,30,193} | {3,144,165} |
{171,111,166} | {180,195,51} | {8,157,189} | {26,7,109} | {200,61,17} |
{197,203,138} | {168,73,123} | {18,185,29} | {160,173,57} | {146,69,143} |
{99,76,53} | {80,15,49} | {66,65,103} | {33,31,90} | {155,60,151} |
{87,177,136} | {125,117,72} | {139,62,100} | {88,120,107} | {44,179,22} |
{162,196,95} | {194,122,71} | {54,198,63} | {134,124,191} | {164,127,110} |
{14,113,58} | {154,119,126} | {116,147,36} | {46,130,37} | {188,67,78} |
{140,48,24} | {2,176,105} | {142,64,128} | {172,10,35} | {174,114,25} |
{145,112,82} | {182,208,94} | {45,38,50} | {56,98,106} | {153,186,28} |
{52,158,27} | {132,92,34} | {206,42,150} | {12,204,202} | {32,16,152} |
{156,74,70} | {170,102,108} | {23,20,96} |
\((5)\ q = 283, g = 17\) | ||||
{208,93} | {164,203} | {0,245} | ||
{39,162,124} | {22,20,66} | {152,264,46} | {44,256,34} | {2,148,80} |
{232,24,116} | {86,56,236} | {238,212,216} | {156,132,260} | {55,122,180} |
{247,114,100} | {281,144,194} | {26,112,49} | {98,92,253} | {158,3,240} |
{121,8,228} | {160,135,214} | {41,74,184} | {133,52,68} | {99,48,6} |
{198,72,217} | {128,129,262} | {134,233,230} | {16,136,145} | {36,278,127} |
{150,28,95} | {119,202,50} | {237,196,252} | {268,267,76} | {276,258,7} |
{27,186,102} | {130,185,42} | {105,280,30} | {45,88,96} | {218,60,177} |
{192,275,54} | {10,118,83} | {154,97,120} | {210,174,101} | {117,246,234} |
{40,140,235} | {250,151,170} | {4,29,32} | {204,138,241} | {200,193,58} |
{274,221,106} | {224,176,63} | {242,195,222} | {178,109,126} | {272,108,115} |
{70,159,168} | {248,172,77} | {137,18,90} | {38,65,91} | {265,183,12} |
{78,89,179} | {1,94,273} | {111,167,188} | {243,197,182} | {226,21,107} |
{189,110,9} | {81,146,181} | {190,223,207} | {147,244,19} | {14,57,191} |
{79,249,220} | {261,266,227} | {199,270,259} | {53,213,142} | {84,169,71} |
{123,5,64} | {205,209,254} | {166,263,271} | {131,103,82} | {255,23,104} |
{171,31,62} | {269,211,206} | {165,59,143} | {149,201,69} | {187,173,15} |
{239,175,219} | {139,215,85} | {47,251,35} | {229,61,157} | {25,113,161} |
{257,67,141} | {73,43,75} | {277,13,37} | {11,51,155} | {231,163,225} |
{153,33,279} | {87,125,17} |
\((6)\ q=307, g= 43\) | ||||
{2,293} | {130,187} | {0,99} | ||
{57,302,227} | {29,185,93} | {109,229,199} | {101,247,23} | {111,207,275} |
{285,289,145} | {33,233,201} | {245,303,9} | {49,179,231} | {63,163,135} |
{79,167,13} | {197,118,291} | {141,133,232} | {183,263,234} | {297,223,212} |
{158,21,97} | {87,38,77} | {283,140,37} | {98,251,209} | {261,279,82} |
{86,31,5} | {121,34,301} | {88,53,47} | {78,15,149} | {161,3,124} |
{193,155,252} | {173,123,170} | {127,71,138} | {151,154,267} | {146,219,265} |
{139,258,225} | {137,191,32} | {228,85,269} | {96,39,83} | {162,157,177} |
{25,129,160} | {134,215,105} | {119,42,117} | {1,260,259} | {125,66,299} |
{171,59,248} | {230,27,235} | {277,270,55} | {81,36,45} | {100,281,295} |
{11,51,210} | {54,107,221} | {165,189,164} | {273,148,95} | {236,91,113} |
{7,69,244} | {253,65,102} | {271,237,136} | {142,257,52} | {89,73,22} |
{48,175,67} | {40,276,181} | {64,206,241} | {116,305,240} | {242,224,75} |
{300,103,20} | {255,80,166} | {195,214,174} | {122,17,194} | {72,115,18} |
{30,43,216} | {169,256,46} | {220,203,108} | {61,150,126} | {296,294,287} |
{156,249,272} | {204,292,19} | {6,284,143} | {288,205,60} | {190,213,298} |
{24,262,217} | {268,222,153} | {226,188,243} | {198,178,26} | {304,282,211} |
{84,168,147} | {186,202,159} | {114,4,35} | {56,120,112} | {239,90,264} |
{106,12,92} | {290,278,76} | {28,62,176} | {44,104,94} | {200,50,196} |
{68,74,274} | {266,218,144} | {184,8,152} | {110,58,246} | {286,250,10} |
{172,128,254} | {70,208,132} | {41,192,14} | {180,131,238} | {182,16,280} |
\((7)\ q=331, g= 98\) | ||||
{236,93} | {251,288} | {0,7} | ||
{187,264,135} | {45,77,27} | {111,241,141} | {23,67,149} | {231,51,129} |
{271,101,313} | {155,31,47} | {131,37,179} | {121,295,55} | {83,279,151} |
{87,49,171} | {221,53,132} | {85,73,250} | {180,325,147} | {245,113,166} |
{19,273,280} | {282,61,277} | {21,281,144} | {239,217,134} | {314,215,143} |
{298,243,59} | {44,311,265} | {99,323,274} | {152,123,181} | {229,324,249} |
{124,63,3} | {58,233,97} | {283,122,145} | {315,139,66} | {158,205,259} |
{247,28,161} | {276,317,75} | {300,57,197} | {240,91,35} | {301,307,242} |
{95,208,207} | {103,204,41} | {163,329,316} | {299,9,254} | {89,176,303} |
{11,290,267} | {118,211,29} | {185,81,38} | {189,26,165} | {159,64,255} |
{326,71,169} | {194,257,177} | {69,36,133} | {138,153,117} | {125,297,270} |
{65,190,209} | {275,154,261} | {305,18,213} | {291,6,287} | {15,62,43} |
{237,235,294} | {14,319,309} | {302,327,293} | {13,106,223} | {191,48,183} |
{193,234,167} | {150,252,137} | {212,306,173} | {184,5,146} | {50,192,107} |
{246,198,127} | {109,214,320} | {272,199,188} | {94,115,78} | {170,296,269} |
{90,258,289} | {25,162,24} | {100,17,68} | {16,206,33} | {105,232,102} |
{200,218,253} | {39,70,266} | {60,82,321} | {92,203,72} | {285,116,172} |
{195,140,260} | {202,278,263} | {108,322,175} | {2,225,228} | {201,30,318} |
{12,1,182} | {4,119,216} | {142,219,210} | {76,308,157} | {156,104,227} |
{268,74,79} | {164,120,8} | {286,80,22} | {20,84,34} | {244,148,86} |
{238,130,328} | {96,224,248} | {98,220,128} | {112,226,186} | {196,256,284} |
{304,10,160} | {168,178,174} | {46,222,230} | {88,42,54} | {56,136,110} |
{32,262,114} | {292,40,126} | {312,310,52} |
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Zheng, H., Chang, Y. & Zhou, J. Direct constructions of large sets of Kirkman triple systems. Des. Codes Cryptogr. 83, 23–32 (2017). https://doi.org/10.1007/s10623-016-0197-4
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DOI: https://doi.org/10.1007/s10623-016-0197-4