Symmetric designs possessing tactical decompositions

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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

Abstract
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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.
[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), 2^nd 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), 2^nd 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.

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.
[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), 2^nd 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), 2^nd 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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