Derived and residual subspace designs

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

Derived and residual subspace designs

Michael Kiermaier ^1, and Reinhard Laue ^2,

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

Abstract
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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.

Keywords: subspace design, derived design, combinatorial design, residual design, large set., $q$-analog.

Mathematics Subject Classification: Primary 51E20; Secondary 05B05, 05B25, 11Tx.

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), 29-37. doi: 10.1016/0012-365X(93)E0096-M.
[2]	M. Braun, Some new designs over finite fields, Bayreuther Math. Schr., 74 (2005), 58-68.
[3]	M. Braun, T. Etzion, P. R. Östergård, A. Vardy and A. Wassermann, Existence of $q$-analogs of Steiner systems, preprint, arXiv:1304.1462
[4]	M. Braun, A. Kerber and R. Laue, Systematic construction of $q$-analogs of designs, Des. Codes Cryptogr., 34 (2005), 55-70. doi: 10.1007/s10623-003-4194-z.
[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), 195-202. doi: 10.1016/j.jcta.2014.01.008.
[6]	S. Braun, Algorithmen zur computerunterstützten Berechnung von $q$-Analoga kombinatorischer Designs, diploma thesis, Universität Bayreuth, 2009.
[7]	S. Braun, Construction of $q$-analogs of combinatorial designs, presentation at the conference Algebraic Combinatorics and Applications (ALCOMA10), Thurnau, Germany, 2010.
[8]	P. J. Cameron, Generalization of Fisher's inequality to fields with more than one element, in Proc. British Combinat. Conf. 1973, Cambridge University Press, 1974, 9-13. doi: 10.1017/CBO9780511662072.003.
[9]	H. Cohn, Projective geometry over $\mathbb F_1$ and the Gaussian binomial coefficients, Amer. Math. Monthly, 111 (2004), 487-495. doi: 10.2307/4145067.
[10]	L. M. H. E. Driessen, $t$-designs, $t \ge 3$, technical report, Technische Universiteit Eindhoven, 1978.
[11]	A. Fazeli, S. Lovett and A. Vardy, Nontrivial $t$-designs over finite fields exist for all $t$, preprint, arXiv:1306.2088
[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), 239-258.
[13]	T. Itoh, A new family of $2$-designs over $GF(q)$ admitting $SL_m(q^l)$, Geom. Dedicata, 69 (1998), 261-286. doi: 10.1023/A:1005057610394.
[14]	M. Kiermaier and M. O. Pavčević, Intersection numbers for subspace designs, J. Combin. Des., to appear: doi:10.1002/jcd.21403 doi: 10.1002/jcd.21403.
[15]	D. C. van Leijenhorst, Orbits on the projective line, J. Combin. Theory Ser. A, 31 (1981), 146-154. doi: 10.1016/0097-3165(81)90011-X.
[16]	M. Miyakawa, A. Munemasa and S. Yoshiara, On a class of small $2$-designs over GF$(q)$, J. Combin. Des., 3 (1995), 61-77. doi: 10.1002/jcd.3180030108.
[17]	M. Schwartz and T. Etzion, Codes and anticodes in the Grassman graph, J. Combin. Theory Ser. A, 97 (2002), 27-42. doi: 10.1006/jcta.2001.3188.
[18]	H. Suzuki, Five days introduction to the theory of designs, 1989, available online at http://subsite.icu.ac.jp/people/hsuzuki/lecturenote/designtheory.pdf
[19]	H. Suzuki, On the inequalities of $t$-designs over a finite field, European J. Combin., 11 (1990), 601-607. doi: 10.1016/S0195-6698(13)80045-5.
[20]	H. Suzuki, $2$-designs over $GF(2^m)$, Graphs Combin., 6 (1990), 293-296. doi: 10.1007/BF01787580.
[21]	H. Suzuki, $2$-designs over $GF(q)$, Graphs Combin., 8 (1992), 381-389. doi: 10.1007/BF02351594.
[22]	L. Teirlinck, Non-trivial $t$-designs without repeated blocks exist for all $t$, Discrete Math., 65 (1987), 301-311. doi: 10.1016/0012-365X(87)90061-6.
[23]	S. Thomas, Designs over finite fields, Geom. Dedicata, 24 (1987), 237-242. doi: 10.1007/BF00150939.
[24]	J. Tits, Sur les analogues algébriques des groupes semi-simples complexes, in Colloque d'Algébre Supérieure, Librairie Gauthiers-Villars, Paris, 1957, 261-289.
[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), 187-192. doi: 10.1007/BF01193690.

show all references

References:

[1]	S. Ajoodani-Namini and G. B. Khosrovashahi, More on halving the complete designs, Discrete Math., 135 (1994), 29-37. doi: 10.1016/0012-365X(93)E0096-M.
[2]	M. Braun, Some new designs over finite fields, Bayreuther Math. Schr., 74 (2005), 58-68.
[3]	M. Braun, T. Etzion, P. R. Östergård, A. Vardy and A. Wassermann, Existence of $q$-analogs of Steiner systems, preprint, arXiv:1304.1462
[4]	M. Braun, A. Kerber and R. Laue, Systematic construction of $q$-analogs of designs, Des. Codes Cryptogr., 34 (2005), 55-70. doi: 10.1007/s10623-003-4194-z.
[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), 195-202. doi: 10.1016/j.jcta.2014.01.008.
[6]	S. Braun, Algorithmen zur computerunterstützten Berechnung von $q$-Analoga kombinatorischer Designs, diploma thesis, Universität Bayreuth, 2009.
[7]	S. Braun, Construction of $q$-analogs of combinatorial designs, presentation at the conference Algebraic Combinatorics and Applications (ALCOMA10), Thurnau, Germany, 2010.
[8]	P. J. Cameron, Generalization of Fisher's inequality to fields with more than one element, in Proc. British Combinat. Conf. 1973, Cambridge University Press, 1974, 9-13. doi: 10.1017/CBO9780511662072.003.
[9]	H. Cohn, Projective geometry over $\mathbb F_1$ and the Gaussian binomial coefficients, Amer. Math. Monthly, 111 (2004), 487-495. doi: 10.2307/4145067.
[10]	L. M. H. E. Driessen, $t$-designs, $t \ge 3$, technical report, Technische Universiteit Eindhoven, 1978.
[11]	A. Fazeli, S. Lovett and A. Vardy, Nontrivial $t$-designs over finite fields exist for all $t$, preprint, arXiv:1306.2088
[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), 239-258.
[13]	T. Itoh, A new family of $2$-designs over $GF(q)$ admitting $SL_m(q^l)$, Geom. Dedicata, 69 (1998), 261-286. doi: 10.1023/A:1005057610394.
[14]	M. Kiermaier and M. O. Pavčević, Intersection numbers for subspace designs, J. Combin. Des., to appear: doi:10.1002/jcd.21403 doi: 10.1002/jcd.21403.
[15]	D. C. van Leijenhorst, Orbits on the projective line, J. Combin. Theory Ser. A, 31 (1981), 146-154. doi: 10.1016/0097-3165(81)90011-X.
[16]	M. Miyakawa, A. Munemasa and S. Yoshiara, On a class of small $2$-designs over GF$(q)$, J. Combin. Des., 3 (1995), 61-77. doi: 10.1002/jcd.3180030108.
[17]	M. Schwartz and T. Etzion, Codes and anticodes in the Grassman graph, J. Combin. Theory Ser. A, 97 (2002), 27-42. doi: 10.1006/jcta.2001.3188.
[18]	H. Suzuki, Five days introduction to the theory of designs, 1989, available online at http://subsite.icu.ac.jp/people/hsuzuki/lecturenote/designtheory.pdf
[19]	H. Suzuki, On the inequalities of $t$-designs over a finite field, European J. Combin., 11 (1990), 601-607. doi: 10.1016/S0195-6698(13)80045-5.
[20]	H. Suzuki, $2$-designs over $GF(2^m)$, Graphs Combin., 6 (1990), 293-296. doi: 10.1007/BF01787580.
[21]	H. Suzuki, $2$-designs over $GF(q)$, Graphs Combin., 8 (1992), 381-389. doi: 10.1007/BF02351594.
[22]	L. Teirlinck, Non-trivial $t$-designs without repeated blocks exist for all $t$, Discrete Math., 65 (1987), 301-311. doi: 10.1016/0012-365X(87)90061-6.
[23]	S. Thomas, Designs over finite fields, Geom. Dedicata, 24 (1987), 237-242. doi: 10.1007/BF00150939.
[24]	J. Tits, Sur les analogues algébriques des groupes semi-simples complexes, in Colloque d'Algébre Supérieure, Librairie Gauthiers-Villars, Paris, 1957, 261-289.
[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), 187-192. doi: 10.1007/BF01193690.

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