# American Institute of Mathematical Sciences

May  2011, 5(2): 199-208. doi: 10.3934/amc.2011.5.199

## Symmetric designs possessing tactical decompositions

 1 University of Zagreb, Faculty of Electrical Engineering and Computing, Department of Applied Mathematics, Unska 3, 10000 Zagreb, Croatia, Croatia

Received  April 2010 Revised  October 2010 Published  May 2011

The main aim of this paper is to construct symmetric designs with trivial automorphism groups. Being aware of the fact that an exhaustive search for parameters $(36,15,6)$ and $(41,16,6)$ is still impossible, we assume that these designs admit a tactical decomposition which would correspond to an orbit structure achieved under an action of an automorphism of order $3$. This constraint proves to be fruitful and allows us to classify simultaneously those symmetric designs with mentioned parameters which admit an automorphism of order $3$ as well as to construct new designs with a trivial automorphism group.
Citation: Ivica Martinjak, Mario-Osvin Pavčević. Symmetric designs possessing tactical decompositions. Advances in Mathematics of Communications, 2011, 5 (2) : 199-208. doi: 10.3934/amc.2011.5.199
##### References:
 [1] W. O. Alltop, An infinite class of $5$-designs, J. Combin. Theory Ser. A, 12 (1972), 390-395. doi: 10.1016/0097-3165(72)90104-5. [2] I. Bouyukliev, V. Fack and J. Winne, $2$-$(31,15,7)$, $2$-$(35,17,8)$ and $2$-$(36,15,6)$ designs with automorphisms of odd prime order, and their related Hadamard matrices and codes, Des. Codes Crypt., 51 (2009), 105-122. doi: 10.1007/s10623-008-9247-x. [3] D. Crnković and D. Held, Some Menon designs having $U(3,3)$ as an automorphism group, Illinois J. Math., 47 (2003), 129-139. [4] V. Ćepulić, On symmetric block designs $(40,13,4)$ with automorphisms of order $5$, Discrete Math., 128 (1994), 45-60. doi: 10.1016/0012-365X(94)90103-1. [5] D. Held, J. Hrabe de Angelis and M.-O. Pavčević, $PSp_4(3)$ as a symmetric $(36,15,6)$ design, Rend. Sem. Mat. Univ. Padova, 101 (1999), 95-98. [6] D. Held and M.-O. Pavčević, Symmetric $(79,27,9)$ design admitting a faithful action of a Frobenius group of order $39$, Europ. J. Combinatorics, 18 (1997), 409-416. doi: 10.1006/eujc.1996.0103. [7] Y. J. Ionin and T. van Trung, Symmetric designs, in "CRC Handbook of Combinatorial Designs'' (eds. C.J. Colbourn and J.H. Dinitz), 2nd edition, CRC Press, Boca Raton, FL, (2007), 110-124. [8] Z. Janko and T. van Trung, Construction of a new symmetric block design for $(78,22,6)$ with the help of tactical decompositions, J. Combin. Theory Ser. A, 40 (1985), 451-455. doi: 10.1016/0097-3165(85)90107-4. [9] P. Kaski and P. R. J. Östergärd, "Classification Algorithms for Codes and Designs,'' Springer, Berlin, 2006. [10] V. Krčadinac, Steiner $2$-designs $S(2,5,41)$ with automorphisms of order $3$, J. Combin. Math. Combin. Comput., 43 (2002), 83-99. [11] E. Lander, "Symmetric Designs: An Algebraic Approach,'' Cambridge University Press, Cambridge, 1983. doi: 10.1017/CBO9780511662164. [12] R. Mathon and A. Rosa, $2$-$(v,k,\lambda)$ designs of small order, in "CRC Handbook of Combinatorial Designs'' (eds. C.J. Colbourn and J.H. Dinitz), 2nd edition, CRC Press, Boca Raton, FL, (2007), 25-58. [13] B. D. McKay, Nauty user's guide (version 1.5), Technical Report TR-CS-90-02, Dep. Computer Science, Australian National University, 1990.

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##### References:
 [1] W. O. Alltop, An infinite class of $5$-designs, J. Combin. Theory Ser. A, 12 (1972), 390-395. doi: 10.1016/0097-3165(72)90104-5. [2] I. Bouyukliev, V. Fack and J. Winne, $2$-$(31,15,7)$, $2$-$(35,17,8)$ and $2$-$(36,15,6)$ designs with automorphisms of odd prime order, and their related Hadamard matrices and codes, Des. Codes Crypt., 51 (2009), 105-122. doi: 10.1007/s10623-008-9247-x. [3] D. Crnković and D. Held, Some Menon designs having $U(3,3)$ as an automorphism group, Illinois J. Math., 47 (2003), 129-139. [4] V. Ćepulić, On symmetric block designs $(40,13,4)$ with automorphisms of order $5$, Discrete Math., 128 (1994), 45-60. doi: 10.1016/0012-365X(94)90103-1. [5] D. Held, J. Hrabe de Angelis and M.-O. Pavčević, $PSp_4(3)$ as a symmetric $(36,15,6)$ design, Rend. Sem. Mat. Univ. Padova, 101 (1999), 95-98. [6] D. Held and M.-O. Pavčević, Symmetric $(79,27,9)$ design admitting a faithful action of a Frobenius group of order $39$, Europ. J. Combinatorics, 18 (1997), 409-416. doi: 10.1006/eujc.1996.0103. [7] Y. J. Ionin and T. van Trung, Symmetric designs, in "CRC Handbook of Combinatorial Designs'' (eds. C.J. Colbourn and J.H. Dinitz), 2nd edition, CRC Press, Boca Raton, FL, (2007), 110-124. [8] Z. Janko and T. van Trung, Construction of a new symmetric block design for $(78,22,6)$ with the help of tactical decompositions, J. Combin. Theory Ser. A, 40 (1985), 451-455. doi: 10.1016/0097-3165(85)90107-4. [9] P. Kaski and P. R. J. Östergärd, "Classification Algorithms for Codes and Designs,'' Springer, Berlin, 2006. [10] V. Krčadinac, Steiner $2$-designs $S(2,5,41)$ with automorphisms of order $3$, J. Combin. Math. Combin. Comput., 43 (2002), 83-99. [11] E. Lander, "Symmetric Designs: An Algebraic Approach,'' Cambridge University Press, Cambridge, 1983. doi: 10.1017/CBO9780511662164. [12] R. Mathon and A. Rosa, $2$-$(v,k,\lambda)$ designs of small order, in "CRC Handbook of Combinatorial Designs'' (eds. C.J. Colbourn and J.H. Dinitz), 2nd edition, CRC Press, Boca Raton, FL, (2007), 25-58. [13] B. D. McKay, Nauty user's guide (version 1.5), Technical Report TR-CS-90-02, Dep. Computer Science, Australian National University, 1990.
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