Cyclic large sets of Steiner triple systems of order
Author:
Kevin T. Phelps
Journal:
Math. Comp. 55 (1990), 821824, S1
MSC:
Primary 05B07
MathSciNet review:
1035942
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Abstract: We examine a class of large sets of Steiner triple systems of order 15 having an automorphism consisting of two fixed points and a 13cycle. We exhibit all members of this class: there are 256 nonisomorphic systems. We examined these members for initial configurations which could lead to a large set of Steiner quadruple systems of order 16 and established that no large set exists having three fixed points and a 13cycle.
 [1]
S.
Bays, Sur les systèmes cycliques de triples de Steiner
différents pour 𝑁 premier (ou puissance de nombre premier)
de la forme 6 𝑛+1 (suite), Comment. Math. Helv.
3 (1931), no. 1, 22–41 (French). MR
1509423, http://dx.doi.org/10.1007/BF01601801
 [2]
F. N. Cole, L. D. Cummings, and H. S. White, Complete enumeration of triple systems in 15 elements, Proc. Nat. Acad. SCi. U.S.A. 3 (1917), 197199.
 [3]
R.
H. F. Denniston, Some packings with Steiner triple systems,
Discrete Math. 9 (1974), 213–227. MR 0369114
(51 #5350)
 [4]
, Sylvester's problems of the 15 school girls, Discrete Math. 9 (1974), 229233.
 [5]
Marshall
Hall Jr., J.
Dean Swift, and Robert
J. Walker, Uniqueness of the projective plane of
order eight, Math. Tables Aids Comput. 10 (1956), 186–194.
MR
0084142 (18,816d), http://dx.doi.org/10.1090/S00255718195600841420
 [6]
Earl
S. Kramer and Dale
M. Mesner, The possible (impossible) systems of 11 disjoint
𝑆(2,3,13)’s (𝑆(3,4,14)’s) with automorphism of
order 11, Utilitas Math. 7 (1975), 55–58. MR 0371680
(51 #7898)
 [7]
Earl
S. Kramer and Dale
M. Mesner, Intersections among Steiner systems, J.
Combinatorial Theory Ser. A 16 (1974), 273–285. MR 0335296
(49 #78)
 [8]
Jia
Xi Lu, On large sets of disjoint Steiner triple systems. I, J.
Combin. Theory Ser. A 34 (1983), no. 2,
140–146. MR
692824 (84e:05031a), http://dx.doi.org/10.1016/00973165(83)900523
 [9]
Jia
Xi Lu, On large sets of disjoint Steiner triple systems. II,
J. Combin. Theory Ser. A 34 (1983), no. 2,
147–155. MR
692825 (84e:05031b), http://dx.doi.org/10.1016/00973165(83)900535
 [10]
Jia
Xi Lu, On large sets of disjoint Steiner triple systems. III,
J. Combin. Theory Ser. A 34 (1983), no. 2,
156–182. MR
692826 (84e:05031c), http://dx.doi.org/10.1016/00973165(83)900547
 [11]
Jia
Xi Lu, On large sets of disjoint Steiner triple systems. IV, V,
VI, J. Combin. Theory Ser. A 37 (1984), no. 2,
136–163, 164–188, 189–192. MR 757612
(86a:05019), http://dx.doi.org/10.1016/00973165(84)900669
 [12]
, On large sets of disjoint Steiner triple systems. V, J. Combin. Theory Ser. A 37 (1984), 164188.
 [13]
, On large sets of disjoint Steiner triple systems. VI, J. Combin. Theory Ser. A 37 (1984), 189192.
 [14]
, On large sets of disjoint Steiner triple systems. VII, manuscript.
 [15]
R.
A. Mathon, K.
T. Phelps, and A.
Rosa, Small Steiner triple systems and their properties, Ars
Combin. 15 (1983), 3–110. MR 706292
(85e:05027a)
 [16]
Richard
M. Wilson, Some partitions of all triples into Steiner triple
systems, Hypergraph Seminar (Proc. First Working Sem., Ohio State
Univ., Columbus, Ohio, 1972; dedicated to Arnold Ross), Springer, Berlin,
1974, pp. 267–277. Lecture Notes in Math., Vol. 411. MR 0369100
(51 #5336)
 [1]
 S. Bays, Sur les systèmes cycliques de triples de Steiner différents pour N premier (ou puisance de nombre premier) de la forme , IIIII, Comment. Math. Helv. 3 (1931), 2241. MR 1509423
 [2]
 F. N. Cole, L. D. Cummings, and H. S. White, Complete enumeration of triple systems in 15 elements, Proc. Nat. Acad. SCi. U.S.A. 3 (1917), 197199.
 [3]
 R. H. F. Denniston, Some packings with Steiner triple systems, Discrete Math. 9 (1974), 213227. MR 0369114 (51:5350)
 [4]
 , Sylvester's problems of the 15 school girls, Discrete Math. 9 (1974), 229233.
 [5]
 M. Hall, Jr., and J. D. Swift, Determination of Steiner triple systems of order 15, MTAC 10 (1956), 186194. MR 0084142 (18:816d)
 [6]
 E. S. Kramer and D. M. Mesner, The possible (impossible) systems of 11 disjoint 's ('s) with automorphism of order 11, Utilitas Math. 7 (1975), 5558. MR 0371680 (51:7898)
 [7]
 , Intersections among Steiner systems, J. Combin. Theory Ser. A 16 (1974), 273285. MR 0335296 (49:78)
 [8]
 JiaXi Lu, On large sets of disjoint Steiner triple systems. I, J. Combin. Theory Ser. A 34 (1983), 140146. MR 692824 (84e:05031a)
 [9]
 , On large sets of disjoint Steiner triple systems. II, J. Combin. Theory Ser. A 34 (1983), 147155. MR 692825 (84e:05031b)
 [10]
 , On large sets of disjoint Steiner triple systems. III, J. Combin. Theory Ser. A 34 (1983), 156182. MR 692826 (84e:05031c)
 [11]
 , On large sets of disjoint Steiner triple systems. IV, J. Combin. Theory Ser. A 37 (1984), 136163. MR 757612 (86a:05019)
 [12]
 , On large sets of disjoint Steiner triple systems. V, J. Combin. Theory Ser. A 37 (1984), 164188.
 [13]
 , On large sets of disjoint Steiner triple systems. VI, J. Combin. Theory Ser. A 37 (1984), 189192.
 [14]
 , On large sets of disjoint Steiner triple systems. VII, manuscript.
 [15]
 R. A. Mathon, K. T. Phelps, and A. Rosa, Small Steiner systems and their properties, Ars Combin. 15 (1983), 3100. MR 706292 (85e:05027a)
 [16]
 R. M. Wilson, Some partitions of all triples into Steiner triple systems, Hypergraph Seminar, Lecture Notes in Math., vol. 411, Springer, Berlin, 1974, pp. 267277. MR 0369100 (51:5336)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199010359421
PII:
S 00255718(1990)10359421
Article copyright:
© Copyright 1990
American Mathematical Society
