# American Institute of Mathematical Sciences

February  2015, 9(1): 105-115. doi: 10.3934/amc.2015.9.105

## Derived and residual subspace designs

 1 Mathematisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany 2 Institut für Informatik, Universität Bayreuth, D-95440 Bayreuth, Germany

Received  May 2014 Revised  October 2014 Published  February 2015

A generalization of forming derived and residual designs from $t$-designs to subspace designs is proposed. A $q$-analog of a theorem by Tran Van Trung, van Leijenhorst and Driessen is proven, stating that if for some (not necessarily realizable) parameter set the derived and residual parameter set are realizable, the same is true for the reduced parameter set.
As a result, we get the existence of several previously unknown subspace designs. Some consequences are derived for the existence of large sets of subspace designs. Furthermore, it is shown that there is no $q$-analog of the large Witt design.
Citation: Michael Kiermaier, Reinhard Laue. Derived and residual subspace designs. Advances in Mathematics of Communications, 2015, 9 (1) : 105-115. doi: 10.3934/amc.2015.9.105
##### References:
 [1] S. Ajoodani-Namini and G. B. Khosrovashahi, More on halving the complete designs,, Discrete Math., 135 (1994), 29. doi: 10.1016/0012-365X(93)E0096-M. Google Scholar [2] M. Braun, Some new designs over finite fields,, Bayreuther Math. Schr., 74 (2005), 58. Google Scholar [3] M. Braun, T. Etzion, P. R. Östergård, A. Vardy and A. Wassermann, Existence of $q$-analogs of Steiner systems,, preprint, (). Google Scholar [4] M. Braun, A. Kerber and R. Laue, Systematic construction of $q$-analogs of designs,, Des. Codes Cryptogr., 34 (2005), 55. doi: 10.1007/s10623-003-4194-z. Google Scholar [5] M. Braun, A. Kohnert, P. R. Östergård and A. Wassermann, Large sets of $t$-designs over finite fields,, J. Combin. Theory Ser. A, 124 (2014), 195. doi: 10.1016/j.jcta.2014.01.008. Google Scholar [6] S. Braun, Algorithmen zur computerunterstützten Berechnung von $q$-Analoga kombinatorischer Designs,, diploma thesis, (2009). Google Scholar [7] S. Braun, Construction of $q$-analogs of combinatorial designs,, presentation at the conference Algebraic Combinatorics and Applications (ALCOMA10), (2010). Google Scholar [8] P. J. Cameron, Generalization of Fisher's inequality to fields with more than one element,, in Proc. British Combinat. Conf. 1973, (1973), 9. doi: 10.1017/CBO9780511662072.003. Google Scholar [9] H. Cohn, Projective geometry over $\mathbb F_1$ and the Gaussian binomial coefficients,, Amer. Math. Monthly, 111 (2004), 487. doi: 10.2307/4145067. Google Scholar [10] L. M. H. E. Driessen, $t$-designs, $t \ge 3$,, technical report, (1978). Google Scholar [11] A. Fazeli, S. Lovett and A. Vardy, Nontrivial $t$-designs over finite fields exist for all $t$,, preprint, (). Google Scholar [12] J. Goldman and G.-C. Rota, On the foundations of combinatorial theory. IV. Finite vector spaces and Eulerian generating functions,, Stud. Appl. Math., 49 (1970), 239. Google Scholar [13] T. Itoh, A new family of $2$-designs over $GF(q)$ admitting $SL_m(q^l)$,, Geom. Dedicata, 69 (1998), 261. doi: 10.1023/A:1005057610394. Google Scholar [14] M. Kiermaier and M. O. Pavčević, Intersection numbers for subspace designs,, J. Combin. Des., (). doi: 10.1002/jcd.21403. Google Scholar [15] D. C. van Leijenhorst, Orbits on the projective line,, J. Combin. Theory Ser. A, 31 (1981), 146. doi: 10.1016/0097-3165(81)90011-X. Google Scholar [16] M. Miyakawa, A. Munemasa and S. Yoshiara, On a class of small $2$-designs over GF$(q)$,, J. Combin. Des., 3 (1995), 61. doi: 10.1002/jcd.3180030108. Google Scholar [17] M. Schwartz and T. Etzion, Codes and anticodes in the Grassman graph,, J. Combin. Theory Ser. A, 97 (2002), 27. doi: 10.1006/jcta.2001.3188. Google Scholar [18] H. Suzuki, Five days introduction to the theory of designs,, 1989, (). Google Scholar [19] H. Suzuki, On the inequalities of $t$-designs over a finite field,, European J. Combin., 11 (1990), 601. doi: 10.1016/S0195-6698(13)80045-5. Google Scholar [20] H. Suzuki, $2$-designs over $GF(2^m)$,, Graphs Combin., 6 (1990), 293. doi: 10.1007/BF01787580. Google Scholar [21] H. Suzuki, $2$-designs over $GF(q)$,, Graphs Combin., 8 (1992), 381. doi: 10.1007/BF02351594. Google Scholar [22] L. Teirlinck, Non-trivial $t$-designs without repeated blocks exist for all $t$,, Discrete Math., 65 (1987), 301. doi: 10.1016/0012-365X(87)90061-6. Google Scholar [23] S. Thomas, Designs over finite fields,, Geom. Dedicata, 24 (1987), 237. doi: 10.1007/BF00150939. Google Scholar [24] J. Tits, Sur les analogues algébriques des groupes semi-simples complexes,, in Colloque d'Algébre Supérieure, (1957), 261. Google Scholar [25] Tran Van Trung, On the construction of $t$-designs and the existence of some new infinite families of simple $5$-designs,, Arch. Math. (Basel), 47 (1986), 187. doi: 10.1007/BF01193690. Google Scholar

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##### References:
 [1] S. Ajoodani-Namini and G. B. Khosrovashahi, More on halving the complete designs,, Discrete Math., 135 (1994), 29. doi: 10.1016/0012-365X(93)E0096-M. Google Scholar [2] M. Braun, Some new designs over finite fields,, Bayreuther Math. Schr., 74 (2005), 58. Google Scholar [3] M. Braun, T. Etzion, P. R. Östergård, A. Vardy and A. Wassermann, Existence of $q$-analogs of Steiner systems,, preprint, (). Google Scholar [4] M. Braun, A. Kerber and R. Laue, Systematic construction of $q$-analogs of designs,, Des. Codes Cryptogr., 34 (2005), 55. doi: 10.1007/s10623-003-4194-z. Google Scholar [5] M. Braun, A. Kohnert, P. R. Östergård and A. Wassermann, Large sets of $t$-designs over finite fields,, J. Combin. Theory Ser. A, 124 (2014), 195. doi: 10.1016/j.jcta.2014.01.008. Google Scholar [6] S. Braun, Algorithmen zur computerunterstützten Berechnung von $q$-Analoga kombinatorischer Designs,, diploma thesis, (2009). Google Scholar [7] S. Braun, Construction of $q$-analogs of combinatorial designs,, presentation at the conference Algebraic Combinatorics and Applications (ALCOMA10), (2010). Google Scholar [8] P. J. Cameron, Generalization of Fisher's inequality to fields with more than one element,, in Proc. British Combinat. Conf. 1973, (1973), 9. doi: 10.1017/CBO9780511662072.003. Google Scholar [9] H. Cohn, Projective geometry over $\mathbb F_1$ and the Gaussian binomial coefficients,, Amer. Math. Monthly, 111 (2004), 487. doi: 10.2307/4145067. Google Scholar [10] L. M. H. E. Driessen, $t$-designs, $t \ge 3$,, technical report, (1978). Google Scholar [11] A. Fazeli, S. Lovett and A. Vardy, Nontrivial $t$-designs over finite fields exist for all $t$,, preprint, (). Google Scholar [12] J. Goldman and G.-C. Rota, On the foundations of combinatorial theory. IV. Finite vector spaces and Eulerian generating functions,, Stud. Appl. Math., 49 (1970), 239. Google Scholar [13] T. Itoh, A new family of $2$-designs over $GF(q)$ admitting $SL_m(q^l)$,, Geom. Dedicata, 69 (1998), 261. doi: 10.1023/A:1005057610394. Google Scholar [14] M. Kiermaier and M. O. Pavčević, Intersection numbers for subspace designs,, J. Combin. Des., (). doi: 10.1002/jcd.21403. Google Scholar [15] D. C. van Leijenhorst, Orbits on the projective line,, J. Combin. Theory Ser. A, 31 (1981), 146. doi: 10.1016/0097-3165(81)90011-X. Google Scholar [16] M. Miyakawa, A. Munemasa and S. Yoshiara, On a class of small $2$-designs over GF$(q)$,, J. Combin. Des., 3 (1995), 61. doi: 10.1002/jcd.3180030108. Google Scholar [17] M. Schwartz and T. Etzion, Codes and anticodes in the Grassman graph,, J. Combin. Theory Ser. A, 97 (2002), 27. doi: 10.1006/jcta.2001.3188. Google Scholar [18] H. Suzuki, Five days introduction to the theory of designs,, 1989, (). Google Scholar [19] H. Suzuki, On the inequalities of $t$-designs over a finite field,, European J. Combin., 11 (1990), 601. doi: 10.1016/S0195-6698(13)80045-5. Google Scholar [20] H. Suzuki, $2$-designs over $GF(2^m)$,, Graphs Combin., 6 (1990), 293. doi: 10.1007/BF01787580. Google Scholar [21] H. Suzuki, $2$-designs over $GF(q)$,, Graphs Combin., 8 (1992), 381. doi: 10.1007/BF02351594. Google Scholar [22] L. Teirlinck, Non-trivial $t$-designs without repeated blocks exist for all $t$,, Discrete Math., 65 (1987), 301. doi: 10.1016/0012-365X(87)90061-6. Google Scholar [23] S. Thomas, Designs over finite fields,, Geom. Dedicata, 24 (1987), 237. doi: 10.1007/BF00150939. Google Scholar [24] J. Tits, Sur les analogues algébriques des groupes semi-simples complexes,, in Colloque d'Algébre Supérieure, (1957), 261. Google Scholar [25] Tran Van Trung, On the construction of $t$-designs and the existence of some new infinite families of simple $5$-designs,, Arch. Math. (Basel), 47 (1986), 187. doi: 10.1007/BF01193690. Google Scholar
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