Large Sets of q -Analogs of Designs Michael Braun, Michael Kiermaier, Axel Kohnert * , Reinhard Laue Universit¨ at Bayreuth, [email protected]Abstract Joining small Large Sets of t-designs to form large Large Sets of t-designs allows to recursively construct infinite series of t-designs. This concept is generalized from ordinary designs over sets to designs over finite vector spaces, i.e. designs over GF (q), using three types of joins. While there are only very few general constructions of such q-designs known so far, from only one large set in the literature and two new ones in this paper this way many infinite series of Large Sets of q-designs with constant block sizes are derived. Keywords: q-analog, t-design, Large Set, subspace design AMS classifications: Primary 51E20; Secondary 05B05, 05B25, 11Txx * † 11.12.2013 1
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Large Sets of q-Analogs of Designs
Michael Braun, Michael Kiermaier, Axel Kohnert∗, Reinhard LaueUniversitat Bayreuth,[email protected]
Abstract
Joining small Large Sets of t-designs to form large Large Sets oft-designs allows to recursively construct infinite series of t-designs.This concept is generalized from ordinary designs over sets to designsover finite vector spaces, i.e. designs over GF (q), using three typesof joins. While there are only very few general constructions of suchq-designs known so far, from only one large set in the literature andtwo new ones in this paper this way many infinite series of Large Setsof q-designs with constant block sizes are derived.
are known: N.J.A. Sloane, at al. The On-Line Ency-
clopedia of Integer Sequences.
15
1.1 Covering Join Constructions
LS2[3](2, 11, 20):
For i = 5, . . . , 14 let Fi = Vi+1/Vi and
Si =
[Vik1
]2
∗Fi[V/Vi+1
k2
]2
.
11 = 5 + 1 + 5, 20 = v1 + 1 + v2.
{V5} ∗F5 LS2[3](2, 5, 14)
LS2[3](0, 5, 6) ∗F6 LS2[3](1, 5, 13)
LS2[3](1, 5, 7) ∗F7 LS2[3](0, 5, 12)
LS2[3](2, 5, 8) ∗F8
[V115
]2
LS2[3](1, 5, 9) ∗F9 LS2[3](1, 5, 10)
LS2[3](1, 5, 10) ∗F10 LS2[3](1, 5, 9)[V115
]2∗F11 LS2[3](2, 5, 8)
LS2[3](1, 5, 12) ∗F12 LS2[3](1, 5, 7)
LS2[3](1, 5, 13) ∗F13 LS2[3](0, 5, 6)
LS2[3](2, 5, 14) ∗F14 {V5}
All the corresponding Large Sets exist, using residual
Large Sets and those for t = 1.
16
Theorem 1.10. ∀(n ≥ 2)∃LS2[3](2, 11, 6n + 2).
Proof. Use LS2[3](2, 5, 6n + 2) with 6n + 2 = 14, 20, . . .
and their residuals. The additional 1-design Large Sets
also exist by recursion 1.4.
The number of rows grows by 6 of the pattern:
v1 t1 v2 t26r + 5 ∗ 6m + 2 2
6r + 6 0 6m + 1 1
6r + 7 1 6m 0
6r + 8 2 6m− 1 ∗6r + 9 1 6m− 2 1
6r + 10 1 6m− 3 1
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Two new Large Sets: Halvings LSq[2](2, 3, 6), q = 3, 5
Theorem 1.11.
∃LSq[N ](2, 3, 6), 4n− 1 > 2s − 1
=⇒∃LSq[N ](2, 2s − 1, 4n + 2)
Proof. s = 2: Use recursion Theorem 1.4
∃LSq[N ](2, 3, 6) =⇒ ∀n∃LSq[N ](2, 3, 4n + 2).
s = 3
∃LSq[N ](2, 3, 6) =⇒ ∀n∃LSq[N ](2, 7, 4n + 2).
Start with n = 2, use the covering join, 7 = 3 + 3 + 1:
{V3} ∗F3 LSq[N ](2, 3, 10)
LSq[N ](0, 3, 4) ∗F4 LSq[N ](1, 3, 9)
LSq[N ](1, 3, 5) ∗F5 LSq[N ](0, 3, 8)
LSq[N ](2, 3, 6) ∗F6
[V/V7
3
]q[
V73
]q∗F7 LSq[N ](2, 3, 6)
LSq[N ](0, 3, 8) ∗F12 LSq[N ](1, 3, 5)
LSq[N ](1, 3, 9) ∗F13 LSq[N ](0, 3, 4)
LSq[N ](2, 3, 10) ∗F14 {V/V8}
Each
LSq[N ](t1, k, v1) ∗F LSq[N ](t2, k, v2)
for t1, t2 ∈ {0, 1} joins residual Large Sets.
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Larger n: The number of rows grows by 4 of the pat-
tern:
v1 t1 v2 t24r − 1 ∗ 4m + 2 2
4r 0 4m + 1 1
4r + 1 1 4m 0
4r + 2 2 4m− 1 ∗Induction on s only iterates the pattern with the pre-
vious value of s, using
2k + 1 = 2(2s− 1) + 1 = 2s+1− 2 + 1 = 2s+1− 1.
Corollary 1.12. For q = 3, 5 there exist infinite se-
ries of halvings for all k = 2s − 1.
19
THANK YOU for YOUR PATIENCE
20
References
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