American Institute of Mathematical Sciences

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

Symmetric designs possessing tactical decompositions

Ivica Martinjak 1, and  Mario-Osvin Pavčević 1,

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.
Keywords: automorphism, tactical decomposition, deterministic algorithm., Symmetric design.
Mathematics Subject Classification: Primary: 05B0.
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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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. Google Scholar

[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.  Google Scholar

[9]

P. Kaski and P. R. J. Östergärd, "Classification Algorithms for Codes and Designs,'' Springer, Berlin, 2006.  Google Scholar

[10]

V. Krčadinac, Steiner $2$-designs $S(2,5,41)$ with automorphisms of order $3$, J. Combin. Math. Combin. Comput., 43 (2002), 83-99.  Google Scholar

[11]

E. Lander, "Symmetric Designs: An Algebraic Approach,'' Cambridge University Press, Cambridge, 1983. doi: 10.1017/CBO9780511662164.  Google Scholar

[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. Google Scholar

[13]

B. D. McKay, Nauty user's guide (version 1.5), Technical Report TR-CS-90-02, Dep. Computer Science, Australian National University, 1990. Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

D. Crnković and D. Held, Some Menon designs having $U(3,3)$ as an automorphism group, Illinois J. Math., 47 (2003), 129-139.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[9]

P. Kaski and P. R. J. Östergärd, "Classification Algorithms for Codes and Designs,'' Springer, Berlin, 2006.  Google Scholar

[10]

V. Krčadinac, Steiner $2$-designs $S(2,5,41)$ with automorphisms of order $3$, J. Combin. Math. Combin. Comput., 43 (2002), 83-99.  Google Scholar

[11]

E. Lander, "Symmetric Designs: An Algebraic Approach,'' Cambridge University Press, Cambridge, 1983. doi: 10.1017/CBO9780511662164.  Google Scholar

[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. Google Scholar

[13]

B. D. McKay, Nauty user's guide (version 1.5), Technical Report TR-CS-90-02, Dep. Computer Science, Australian National University, 1990. Google Scholar

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