Classification of simple 2-(11, 3, 3) designs

We present an orderly algorithm for classifying triple systems. Subsequently, we show that there exist exactly 7038,699,746 nonisomorphic simple 2-(11,3,3) designs. The method is also used to confirm the previously accomplished classifications of 2-(8,3,6), 2-(12,3,2) and 2-(19,3,1) designs.

[1]  A. S. Hedayat,et al.  On Theory and Applications of BIB Designs with Repeated Blocks. , 1977 .

[2]  Reinhard Laue,et al.  Algorithms for group actions applied to graph generation , 1995, Groups and Computation.

[3]  Brendan D. McKay,et al.  Isomorph-Free Exhaustive Generation , 1998, J. Algorithms.

[4]  Donald E. Knuth,et al.  Dancing links , 2000, cs/0011047.

[5]  Patric R. J. Östergård,et al.  The Steiner triple systems of order 19 , 2004, Math. Comput..

[6]  P. Kaski,et al.  Classification Algorithms for Codes and Designs , 2005 .

[7]  F. N. Cole.  The triad systems of thirteen letters , 1913 .

[8]  R. Read.  Every one a Winner or how to Avoid Isomorphism Search when Cataloguing Combinatorial Configurations , 1978 .

[9]  A. V. Ivanov.  Constructive Enumeration of Incidence Systems , 1985 .

[10]  Marshall Hall,et al.  Determination of Steiner triple systems of order 15 , 1955 .

[11]  A. S. Hedayat,et al.  BIB(8, 56, 21, 3, 6) and BIB(10, 30, 9, 3, 2) Designs with Repeated Blocks , 1984, J. Comb. Theory A.

[12]  F N Cole,et al.  The Complete Enumeration of Triad Systems in 15 Elements. , 1917, Proceedings of the National Academy of Sciences of the United States of America.

[13]  Patric R. J. Östergård.  Enumeration of 2-(12, 3, 2) designs , 2000, Australas. J Comb..

[14]  Solomon W. Golomb,et al.  Backtrack Programming , 1965, JACM.

[15]  Jim Davies,et al.  Millennial Perspectives in Computer Science , 2000 .

[16]  Vladimir D. Tonchev,et al.  Results on the support of BIB designs , 1989 .

[17]  Rufus Walker,et al.  An enumerative technique for a class of combinatorial problems , 1960 .

[18]  Gholamreza B. Khosrovshahi,et al.  Classification of 6-(14,7,4) designs with nontrivial automorphism groups , 2002 .

[19]  Louise D. Cummings,et al.  Complete classification of the triad systems on fifteen elements , 1919 .

[20]  Douglas R Stinson,et al.  Steiner Triple Systems of Order 19 Contain a Subsystem of Order 9 By , 2010 .

[21]  Charles J. Colbourn,et al.  Steiner triple systems of order 19 with nontrivial automorphism group , 1992 .

[22]  Rudolf Mathon,et al.  A census of 2-(9, 3, 3) designs , 1992, Australas. J Comb..

[23]  B. Tayfeh-Rezaie,et al.  Backtracking algorithm for finding t‐designs , 2003 .

[24]  C. Colbourn,et al.  Handbook of Combinatorial Designs , 2006 .

[25]  Patric R. J. Östergård,et al.  Computational Methods in Design Theory , 2006 .

[26]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .