# 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.  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.  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.   Google Scholar [4] V. Ćepulić, On symmetric block designs $(40,13,4)$ with automorphisms of order $5$,, Discrete Math., 128 (1994), 45.  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.   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.  doi: 10.1006/eujc.1996.0103.  Google Scholar [7] Y. J. Ionin and T. van Trung, Symmetric designs,, in, (2007), 110.   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.  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, (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.   Google Scholar [11] E. Lander, "Symmetric Designs: An Algebraic Approach,'', Cambridge University Press, (1983).  doi: 10.1017/CBO9780511662164.  Google Scholar [12] R. Mathon and A. Rosa, $2$-$(v,k,\lambda)$ designs of small order,, in, (2007), 25.   Google Scholar [13] B. D. McKay, Nauty user's guide (version 1.5),, Technical Report TR-CS-90-02, (1990), 90.   Google Scholar

show all references

##### References:
 [1] W. O. Alltop, An infinite class of $5$-designs,, J. Combin. Theory Ser. A, 12 (1972), 390.  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.  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.   Google Scholar [4] V. Ćepulić, On symmetric block designs $(40,13,4)$ with automorphisms of order $5$,, Discrete Math., 128 (1994), 45.  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.   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.  doi: 10.1006/eujc.1996.0103.  Google Scholar [7] Y. J. Ionin and T. van Trung, Symmetric designs,, in, (2007), 110.   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.  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, (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.   Google Scholar [11] E. Lander, "Symmetric Designs: An Algebraic Approach,'', Cambridge University Press, (1983).  doi: 10.1017/CBO9780511662164.  Google Scholar [12] R. Mathon and A. Rosa, $2$-$(v,k,\lambda)$ designs of small order,, in, (2007), 25.   Google Scholar [13] B. D. McKay, Nauty user's guide (version 1.5),, Technical Report TR-CS-90-02, (1990), 90.   Google Scholar
 [1] Giuseppe Geymonat, Françoise Krasucki. Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains. Communications on Pure & Applied Analysis, 2009, 8 (1) : 295-309. doi: 10.3934/cpaa.2009.8.295 [2] Nguyen Van Thoai. Decomposition branch and bound algorithm for optimization problems over efficient sets. Journal of Industrial & Management Optimization, 2008, 4 (4) : 647-660. doi: 10.3934/jimo.2008.4.647 [3] Rongliang Chen, Jizu Huang, Xiao-Chuan Cai. A parallel domain decomposition algorithm for large scale image denoising. Inverse Problems & Imaging, 2019, 13 (6) : 1259-1282. doi: 10.3934/ipi.2019055 [4] Harish Garg. Solving structural engineering design optimization problems using an artificial bee colony algorithm. Journal of Industrial & Management Optimization, 2014, 10 (3) : 777-794. doi: 10.3934/jimo.2014.10.777 [5] Van Cyr, John Franks, Bryna Kra, Samuel Petite. Distortion and the automorphism group of a shift. Journal of Modern Dynamics, 2018, 13: 147-161. doi: 10.3934/jmd.2018015 [6] Behrouz Kheirfam. A full Nesterov-Todd step infeasible interior-point algorithm for symmetric optimization based on a specific kernel function. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 601-614. doi: 10.3934/naco.2013.3.601 [7] Yanqin Bai, Lipu Zhang. A full-Newton step interior-point algorithm for symmetric cone convex quadratic optimization. Journal of Industrial & Management Optimization, 2011, 7 (4) : 891-906. doi: 10.3934/jimo.2011.7.891 [8] Arvind Ayyer, Carlangelo Liverani, Mikko Stenlund. Quenched CLT for random toral automorphism. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 331-348. doi: 10.3934/dcds.2009.24.331 [9] Xiao-Hong Liu, Wei-Zhe Gu. Smoothing Newton algorithm based on a regularized one-parametric class of smoothing functions for generalized complementarity problems over symmetric cones. Journal of Industrial & Management Optimization, 2010, 6 (2) : 363-380. doi: 10.3934/jimo.2010.6.363 [10] Esa Järvenpää, Maarit Järvenpää, R. Daniel Mauldin. Deterministic and random aspects of porosities. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 121-136. doi: 10.3934/dcds.2002.8.121 [11] Vsevolod Laptev. Deterministic homogenization for media with barriers. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 29-44. doi: 10.3934/dcdss.2015.8.29 [12] Van Cyr, Bryna Kra. The automorphism group of a minimal shift of stretched exponential growth. Journal of Modern Dynamics, 2016, 10: 483-495. doi: 10.3934/jmd.2016.10.483 [13] Nikolay Yankov, Damyan Anev, Müberra Gürel. Self-dual codes with an automorphism of order 13. Advances in Mathematics of Communications, 2017, 11 (3) : 635-645. doi: 10.3934/amc.2017047 [14] Paweł Góra, Abraham Boyarsky, Zhenyang LI, Harald Proppe. Statistical and deterministic dynamics of maps with memory. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4347-4378. doi: 10.3934/dcds.2017186 [15] Leonid A. Bunimovich, Alex Yurchenko. Deterministic walks in rigid environments with aging. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 37-46. doi: 10.3934/dcdsb.2008.9.37 [16] Oliver Knill. A deterministic displacement theorem for Poisson processes. Electronic Research Announcements, 1997, 3: 110-113. [17] Fabio Augusto Milner, Ruijun Zhao. A deterministic model of schistosomiasis with spatial structure. Mathematical Biosciences & Engineering, 2008, 5 (3) : 505-522. doi: 10.3934/mbe.2008.5.505 [18] Siwei Yu, Jianwei Ma, Stanley Osher. Geometric mode decomposition. Inverse Problems & Imaging, 2018, 12 (4) : 831-852. doi: 10.3934/ipi.2018035 [19] Stefano Bianchini, Daniela Tonon. A decomposition theorem for $BV$ functions. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1549-1566. doi: 10.3934/cpaa.2011.10.1549 [20] Dino Festi, Alice Garbagnati, Bert Van Geemen, Ronald Van Luijk. The Cayley-Oguiso automorphism of positive entropy on a K3 surface. Journal of Modern Dynamics, 2013, 7 (1) : 75-97. doi: 10.3934/jmd.2013.7.75

2018 Impact Factor: 0.879