Derived and Residual Subspace Designs

Transcription

1 Derived and Residual Subspace Designs Michael Kiermaier, Reinhard Laue Universität Bayreuth, Abstract A generalization of forming derived and residual designs from t- designs to subspace designs is proposed. A q-analog of a Theorem by van Tran, 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. Furthermore, some consequences are derived for the existence of large sets of subspace designs. Keywords: q-analog, t-design, large set, subspace design AMS classifications: Primary 51E20; Secondary 05B05, 05B25, 11Txx 1

2 1 Classic and Subspace t-designs t-(v, k, λ) designd = (, B) t-(v, k, λ) q designd = (, B) point set size v GF (q) vector space dim v B ( ) k B [ ] k q each T ( ) [ t in λ B B each T ] t q in λ B B Subspace designs: Definition: Cameron 1974 [8], first found for t 2: Thomas 1987 [14]. Introduction to theory: Suzuki, day 4 in [12, Day 4]. New: t = 2 and t = 3 M.,S.Braun et al. [2, 3, 7, 4], 2-(13, 3, 1) 2 in Braun,Etzion,Östergard,Wassermann [5], q-analog to Teirlinck s theorem [13]: t simple t- subspace design: Fazelli,Lovett,ardi [10]. 2

3 Analog of classic basic theory still missing. Important tools: Derived, residual designs. D = (, B) a t-(v, k, λ) design, x Red(D) (, B) reduced (t 1)-(v, k, λ v t+1 k t+1 ) design Der x (D) ( \ {x}, {B \ {x} : x B B}) derived (t 1)-(v 1, k 1, λ) design Res x (D) ( \ {x}, {B B : x B}) residual v k (t 1)-(v 1, k, λ k t+1 ) design D = (, B) a t-(v, k, λ) q design, U [ 1 ]q, H [ ] v 1 Red(D) (, B) reduced (t 1)-(v, k, λ qv t+1 1 q k t+1 1 q design Der U (D) (/U, {B/U : B B, U B}) derived (t 1)-(v 1, k 1, λ) q design Res H (D) (H, {B : B B, B H}) q residual in H (t 1)-(v 1, k, λ 1 q k t+1 1 q design If q 1 then q-version set-version. q. 3

4 If D exists then also Red(D), Red(Red(D)),... q So, all indices λ, λ v k 1,... must be integers. q k (t 1) 1 t-(v, k, λ) q is admissible if all these values are integers. Main Problem: Which admissible parameter sets are realizable? Example: 3-(22, 6, 1) q, 4-(23, 7, 1) q, 5-(24, 8, 1) q are not admissible. 2-(7, 3, 1) q admissible, Open: 2-(7, 3, 1) q realizable? 4

5 Theorem 1.1. Let 0 < t < k < v t-(v, k, λ) q design D = Red(D), Der U (D), Res H (D). Proof. Red(D), Der U (D): straightforward computation. Res H (D): follows from Lemma 4.2 for dual designs in Suzuki [12]. t-(v, k, λ) design D Red(D) Der U (D) Res H (D) Der U (D) is factor design. Res H (D) is subdesign. 5

6 2 A Construction Theorem Analog of a classic theorem of van Tran [16], van Leijenhorst [11], Driessen [9]. Theorem 2.1. t-(v, k, λ) q parameter set whose derived and residual parameter sets are realizable. Then its reduced parameter set is realizable, too. red (t 1)-(v, k, λ qv t+1 1 q k t+1 1 ) q t-(v, k, λ) q der (t 1)-(v 1, k 1, λ) q res (t 1)-(v 1, k, λ q v k 1 q k t+1 1 ) q 6

7 Proof. a GF (q) vector space of dim v, Construct D = (, B) with parameters of reduced design: U 1-dim subspace of. Assume designs on /U: D = (/U, B ) derived B/U B gives block B B that contains U B k 1 U 1 {0} D = (/U, B ) residual B/U B All complements K of U in B are blocks in B. U 1 {0} B k k K q k such blocks K for each B/U B. 7

8 Each t 1-subspace T in λ qv t+1 1 q k t+1 1 U T = U B, dim(t/u) = t 2 T/U in λ U T B [ (v 1) (t 2) (t 1) (t 2) [ (k 1) (t 2) (t 1) (t 2) = ] a) dim(b/u) = k 1, derived U {0} ] q q blocks of B: = λ qv t+1 1 q k t+1 1 blocks B k-1 t-1 T T in λ such blocks B b) dim(b + U/U) = k, residual U {0} T in q k t+1 λ B+U k t-1 T B q v k 1 q k t+1 1 such blocks The sum of a) and b) evaluates to right term. 8

9 Remark: Construction need not reproduce Red(D) if D exists! 2-(13, 3, 1) 2 has blocks B 1, B 2 not containing 1-dim subspace U. If U + B 1 = U + B 2 then dim(b 1 B 2 ) = 2. So, B 1 B 2 would lie in 2 blocks. 9

10 Table 1: Parameter sets of subspace designs unknown exists by exists by new t-(v, k, λ) q derived source residual source reduced 3-(8, 4, 3) 2 2-(7, 3, 3) 2 [4] 2-(7, 4, 15) 2 [4] 2-(8, 4, 63) 2 3-(8, 4, 4) 2 2-(7, 3, 4) 2 [4] 2-(7, 4, 20) 2 [4] 2-(8, 4, 84) 2 3-(8, 4, 7) 2 2-(7, 3, 7) 2 [4] 2-(7, 4, 35) 2 [4] 2-(8, 4, 147) 2 3-(8, 4, 8) 2 2-(7, 3, 8) 2 [4] 2-(7, 4, 40) 2 [4] 2-(8, 4, 168) 2 3-(8, 4, 9) 2 2-(7, 3, 9) 2 [4] 2-(7, 4, 45) 2 [4] 2-(8, 4, 189) 2 3-(8, 4, 12) 2 2-(7, 3, 12) 2 [4] 2-(7, 4, 60) 2 [4] 2-(8, 4, 252) 2 3-(8, 4, 13) 2 2-(7, 3, 13) 2 [4] 2-(7, 4, 65) 2 [4] 2-(8, 4, 273) 2 3-(8, 4, 14) 2 2-(7, 3, 14) 2 [4] 2-(7, 4, 70) 2 [4] 2-(8, 4, 294) 2 3-(10, 4, 21) 2 2-(9, 3, 21) 2 [7] 2-(9, 4, 441) 2 [7] 2-(10, 4, 1785) 2 3-(10, 4, 22) 2 2-(9, 3, 22) 2 [4] 2-(9, 4, 462) 2 [4] 2-(10, 4, 1870) 2 3-(10, 4, 42) 2 2-(9, 3, 42) 2 [7] 2-(9, 4, 882) 2 [7] 2-(10, 4, 3570) 2 3-(10, 4, 43) 2 2-(9, 3, 43) 2 [4] 2-(9, 4, 903) 2 [4] 2-(10, 4, 3655) 2 3-(10, 4, 63) 2 2-(9, 3, 63) 2 [7] 2-(9, 4, 1323) 2 [7] 2-(10, 4, 5355) 2 10

11 3 Application to large sets Definition 3.1. Large set LS q [N](t, k, v): Partition of [ ] k q into N t-(v, k, λ) q designs. Theorem 3.1. [6, 12] t 1 LS q [N](t, k, v) LS q [N](t, v k, v) Theorem 3.2. t 1 LS q [N](t, k, v) = LS q [N](t 1, k 1, v 1), LS q [N](t 1, k, v 1) Theorem 2.1 applied to pairs of derived and residual designs from large sets yields a new large set. Corollary 3.3 (q-analog of [1, Lemma. 4]). LS q [N](t, k 1, v 1), LS q [N](t, k, v 1) = LS q [N](t, k, v) For t 2 such combinable pairs of large sets have not been found so far. There are LS 2 [3](2, 3, 8) and LS 2 [3](2, 5, 8), see [6]. If an LS 2 [3](2, 4, 8) could be found then Corollary 3.3 would imply the existence of large sets with the parameters LS 2 [3](2, 4, 9), LS 2 [3](2, 5, 9) and LS 2 [3](2, 5, 10). Acknowledgement The authors thank Thomas Feulner for stimulating discussions on this topic. 11

12 References [1] Ajoodani-Namini, S. and Khosrovashahi, G.B., More on halving the complete designs, Discrete Mathematics 135 (1994), [2] Braun, Michael, Kerber, Adalbert and Laue, Reinhard, Systematic construction of q-analogs of designs, Designs, Codes, Cryptography 34, (2005), [3] Braun, Michael, Some new designs over finite fields, Bayreuther Math. Schr. 74 (2005), [4] Braun, Michael, Designs over finite fields - revisited Fq10, Ghent (2011). [5] Braun, Michael, Etzion, Tuvi, Östergard, Patric R., ardi, Alexander and Wassermann, Alfred, Existence of q-analogs of Steiner systems, ArXiv: v2. [6] Braun, Michael, Kohnert, Axel, Östergard, Patric R. and Wassermann, Alfred, Large sets of t-designs over finite fields ArXiv: v1. [7] Braun, Stefanie, Construction of q-analogs of combinatorial designs, Alcoma 10 Thurnau, [8] Cameron, P. J., Generalization of Fisher s inequality to fields with more than one element. In: Combinatorics. Proceedings of the British Combinatorial Conference 1973, London Mathematical Society Lecture Note Series 13 (1974), 9 13, Cambridge. [9] Driessen, Leon M.H.E., t-designs, t 3, Technical Report, Department of Mathematics, Eindhoven University of Technology, (1978). [10] Fazeli, Arman, Lovett, Shachar and ardi, Alexander, Nontrivial t-designs over finite fields exist for all t, preprint. [11] van Leijenhorst, Dirk C., Orbits on the projective line, J. Comb. Theory, Ser. A 31 (1981),

13 [12] Suzuki, Hiroshi Five Days Introduction to the Theory of Designs, 1989, available online at [13] Tierlinck, Luc, Non-trivial t-designs without repeated blocks exist for all t, Discrete Math. 65 (1987), [14] Thomas, Simon, Designs over finite fields, Geom. Dedicata 24 (1987), [15] Tits, Jacques, Sur les analogues algébriques des groupes semi-simples complexes, in Colloque d Algébre Supérieure, tenue à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathèmatiques Ètablissements Ceuterick, Louvain, Paris: Librairie Gauthiers-illars, (1957), [16] 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),