# American Institute of Mathematical Sciences

August  2016, 10(3): 511-524. doi: 10.3934/amc.2016022

## Construction of 3-designs using $(1,\sigma)$-resolution

 1 Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany

Received  May 2015 Revised  November 2015 Published  August 2016

The paper deals with recursive constructions for simple 3-designs based on other 3-designs having $(1, \sigma)$-resolution. The concept of $(1, \sigma)$-resolution may be viewed as a generalization of the parallelism for designs. We show the constructions and their applications to produce many previously unknown infinite families of simple 3-designs. We also include a discussion of $(1,\sigma)$-resolvability of the constructed designs.
Citation: Tran van Trung. Construction of 3-designs using $(1,\sigma)$-resolution. Advances in Mathematics of Communications, 2016, 10 (3) : 511-524. doi: 10.3934/amc.2016022
##### References:
 [1] R. D. Baker, Partitioning the planes of $AG_{2m}(2)$ into 2-designs,, Discr. Math., 15 (1976), 205. [2] Z. Baranyai, On the factorization of the complete uniform hypergraph,, in Proc. Erdös-Colloquium Keszthely, (1973), 91. [3] T. Beth, D. Jungnickel and H. Lenz, Design Theory,, Cambridge Univ. Press, (1986). [4] J. Bierbrauer, Some friends of Alltop's designs $4-(2^f+1,5,5)$,, J. Combin. Math. Combin. Comput., 36 (2001), 43. [5] J. Bierbrauer and T. van Trung, Shadow and shade of designs $4-(2^f+1,6,10)$,, unpublished manuscript, (1994). [6] R. C. Bose, A note on the resolvability of balanced incomplete block designs,, Sankhyā, 6 (1942), 105. [7] C. J. Colbourn and J. H. Dinitz, The CRC Handbook of Combinatorial Designs,, CRC Press, (1996). doi: 10.1201/9781420049954. [8] L. H. M. E. Driessen, $t$-designs, $t\geq 3$, Technical Report,, Dep. Math., (1978). [9] M. Jimbo, Y. Kunihara, R. Laue and M. Sawa, Unifying some known infinite families of combinatorial 3-designs,, J. Combin. Theory Ser. A, 118 (2011), 1072. doi: 10.1016/j.jcta.2010.10.007. [10] D. Jungnickel and S. A. Vanstone, On resolvable designs $S_3(3;4,v)$,, J. Combin. Theory A, 43 (1986), 334. doi: 10.1016/0097-3165(86)90073-7. [11] R. Laue, Resolvable $t$-designs,, Des. Codes Cryptogr., 32 (2004), 277. doi: 10.1023/B:DESI.0000029230.50742.8f. [12] K. T. Phelps, D. R. Stinson and S. A. Vanstone, The existence of simple $S_3(3,4,v)$,, Discrete Math., 77 (1989), 255. doi: 10.1016/0012-365X(89)90364-6. [13] S. S. Shrikhande and D. Raghavarao, A method of construction of incomplete block designs,, Sankhyā A, 25 (1963), 399. [14] S. S. Shrikhande and D. Raghavarao, Affine $\alpha$-resolvable incomplete block designs,, in Contributions to Statistics, (1963), 471. [15] D. R. Stinson, C. M. Swanson and T. van Trung, A new look at an old construction: Constructing (simple) 3-designs from resolvable 2-designs,, Discrete Math., 325 (2014), 23. doi: 10.1016/j.disc.2014.02.009. [16] T. van Trung, Recursive constructions for 3-designs and resolvable 3-designs,, J. Statist. Plann. Inference, 95 (2001), 341. doi: 10.1016/S0378-3758(00)00308-6. [17] T. van Trung, Construction of 3-designs using parallelism,, J. Geom., 67 (2000), 223. doi: 10.1007/BF01220313.

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##### References:
 [1] R. D. Baker, Partitioning the planes of $AG_{2m}(2)$ into 2-designs,, Discr. Math., 15 (1976), 205. [2] Z. Baranyai, On the factorization of the complete uniform hypergraph,, in Proc. Erdös-Colloquium Keszthely, (1973), 91. [3] T. Beth, D. Jungnickel and H. Lenz, Design Theory,, Cambridge Univ. Press, (1986). [4] J. Bierbrauer, Some friends of Alltop's designs $4-(2^f+1,5,5)$,, J. Combin. Math. Combin. Comput., 36 (2001), 43. [5] J. Bierbrauer and T. van Trung, Shadow and shade of designs $4-(2^f+1,6,10)$,, unpublished manuscript, (1994). [6] R. C. Bose, A note on the resolvability of balanced incomplete block designs,, Sankhyā, 6 (1942), 105. [7] C. J. Colbourn and J. H. Dinitz, The CRC Handbook of Combinatorial Designs,, CRC Press, (1996). doi: 10.1201/9781420049954. [8] L. H. M. E. Driessen, $t$-designs, $t\geq 3$, Technical Report,, Dep. Math., (1978). [9] M. Jimbo, Y. Kunihara, R. Laue and M. Sawa, Unifying some known infinite families of combinatorial 3-designs,, J. Combin. Theory Ser. A, 118 (2011), 1072. doi: 10.1016/j.jcta.2010.10.007. [10] D. Jungnickel and S. A. Vanstone, On resolvable designs $S_3(3;4,v)$,, J. Combin. Theory A, 43 (1986), 334. doi: 10.1016/0097-3165(86)90073-7. [11] R. Laue, Resolvable $t$-designs,, Des. Codes Cryptogr., 32 (2004), 277. doi: 10.1023/B:DESI.0000029230.50742.8f. [12] K. T. Phelps, D. R. Stinson and S. A. Vanstone, The existence of simple $S_3(3,4,v)$,, Discrete Math., 77 (1989), 255. doi: 10.1016/0012-365X(89)90364-6. [13] S. S. Shrikhande and D. Raghavarao, A method of construction of incomplete block designs,, Sankhyā A, 25 (1963), 399. [14] S. S. Shrikhande and D. Raghavarao, Affine $\alpha$-resolvable incomplete block designs,, in Contributions to Statistics, (1963), 471. [15] D. R. Stinson, C. M. Swanson and T. van Trung, A new look at an old construction: Constructing (simple) 3-designs from resolvable 2-designs,, Discrete Math., 325 (2014), 23. doi: 10.1016/j.disc.2014.02.009. [16] T. van Trung, Recursive constructions for 3-designs and resolvable 3-designs,, J. Statist. Plann. Inference, 95 (2001), 341. doi: 10.1016/S0378-3758(00)00308-6. [17] T. van Trung, Construction of 3-designs using parallelism,, J. Geom., 67 (2000), 223. doi: 10.1007/BF01220313.
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