CONSTRUCTION OF SIMPLE 3-DESIGNS USING RESOLUTION

Transcription

1 CONSTRUCTION OF SIMPLE 3-DESIGNS USING RESOLUTION Tran van Trung Institut für Experimentelle Mathematik Universität Duisburg-Essen ALCOMA15 Kloster Banz, Germany March 15 20, 2015

2 OUTLINE Generic constructions Applications (1, σ)-resolvable 3-designs

3 Definition A t (v, k, λ)-design (X, B) is said to be (s, σ)-resolvable if its block set B can be partitioned into w classes π 1,..., π w such that (X, π i ) is a s (v, k, σ) design for all i = 1,..., w, where 1 s t. Each π i is called a resolution class

4 Definition A t (v, k, λ)-design (X, B) is said to be (s, σ)-resolvable if its block set B can be partitioned into w classes π 1,..., π w such that (X, π i ) is a s (v, k, σ) design for all i = 1,..., w, where 1 s t. Each π i is called a resolution class Definition Let D be a t (v, k, λ) design (D may have repeated blocks) admitting a (s, σ)-resolution with π 1,..., π w as resolution classes. Define a distance between any two classes π i and π j by d(π i, π j ) = min{ i j, w i j }.

5 n 1, integer. {k 1,..., k n, k n+1,..., k 2n } and k, integers, such that 2 k 1 <... < k n < k/2 and k i + k n+i = k for i = 1,..., n.

6 n 1, integer. {k 1,..., k n, k n+1,..., k 2n } and k, integers, such that 2 k 1 <... < k n < k/2 and k i + k n+i = k for i = 1,..., n. Assume there exist 2n 3-designs D i = (X, B i ) with parameters 3 (v, k i, λ (i) ) having a (1, σ (i) )-resolution such that w i = w n+i for all i = 1,..., n, where w j is the number of (1, σ (j) )-resolution classes of D j.

7 n 1, integer. {k 1,..., k n, k n+1,..., k 2n } and k, integers, such that 2 k 1 <... < k n < k/2 and k i + k n+i = k for i = 1,..., n. Assume there exist 2n 3-designs D i = (X, B i ) with parameters 3 (v, k i, λ (i) ) having a (1, σ (i) )-resolution such that w i = w n+i for all i = 1,..., n, where w j is the number of (1, σ (j) )-resolution classes of D j. Also assume that 1 For each pair (D i, D n+i ), 1 i n, either D i or D n+i has to be simple.

8 n 1, integer. {k 1,..., k n, k n+1,..., k 2n } and k, integers, such that 2 k 1 <... < k n < k/2 and k i + k n+i = k for i = 1,..., n. Assume there exist 2n 3-designs D i = (X, B i ) with parameters 3 (v, k i, λ (i) ) having a (1, σ (i) )-resolution such that w i = w n+i for all i = 1,..., n, where w j is the number of (1, σ (j) )-resolution classes of D j. Also assume that 1 For each pair (D i, D n+i ), 1 i n, either D i or D n+i has to be simple. 2 If a D j, j {i, n + i}, is not simple, then D j is a union of a j copies of a simple 3 (v, k j, α (j) ) design C j, wherein C j admits a (1, σ (j) )-resolution. Thus, λ (j) = a j α (j).

9 If D j is not simple, (i.e. D j is a union of a j copies of a simple 3 (v, k j, α (j) ) design C j, where P (j) = {π (j) 1,..., π(j) t } j is a (1, σ (j) )-resolution of C j ), then the corresponding (1, σ (j) )-resolution of D j is the concatenation of a j sets P (j). So, the w j = a j t j resolution classes of D j are of the form π (j) 1,..., π(j) t j, π (j) 1,..., π(j) t j,..., π (j) 1,..., π(j) t j

10 If D j is not simple, (i.e. D j is a union of a j copies of a simple 3 (v, k j, α (j) ) design C j, where P (j) = {π (j) 1,..., π(j) t } j is a (1, σ (j) )-resolution of C j ), then the corresponding (1, σ (j) )-resolution of D j is the concatenation of a j sets P (j). So, the w j = a j t j resolution classes of D j are of the form π (j) 1,..., π(j) t j, π (j) 1,..., π(j) t j,..., π (j) 1,..., π(j) t j If k 1 = 2, then D 1 is a union of a 1 copies of the trivial 2 (v, 2, 1) design i.e. D 1 is considered as a 3-design with λ (1) = 0.

11 If D j is not simple, (i.e. D j is a union of a j copies of a simple 3 (v, k j, α (j) ) design C j, where P (j) = {π (j) 1,..., π(j) t } j is a (1, σ (j) )-resolution of C j ), then the corresponding (1, σ (j) )-resolution of D j is the concatenation of a j sets P (j). So, the w j = a j t j resolution classes of D j are of the form π (j) 1,..., π(j) t j, π (j) 1,..., π(j) t j,..., π (j) 1,..., π(j) t j If k 1 = 2, then D 1 is a union of a 1 copies of the trivial 2 (v, 2, 1) design i.e. D 1 is considered as a 3-design with λ (1) = 0. If necessary, also assume that there exists a 3 (v, k, Λ) design D = (X, B).

12 Notation: π (l) 1,..., π(l) w l : the w l classes in a (1, σ (l) )-resolution of D l, l = 1,..., 2n. Recall that w n+h = w h. The distance defined on the classes of D l is then d (l) (π (l), π (l) ) = min{ i j, w i j l i j }. b (j) = σ (j) v/k : the number of blocks in each resolution class of of D j. u j = σ (j) : the number of blocks containing a point in each resolution class of of D j. λ (j) 2 = λ(j) (v 2)/(k j 2) : the number of blocks of D j containing two points.

13 Construction I Let D i = ( X, B i ) be a copy of D i defined on X such that X X =. Also let D = ( X, B) be a copy of D. Define blocks on the point set X X as follows: I. blocks of D and blocks of D; II. blocks of the form A B for any pair A π (h) and i B π (n+h) with ε j h d (h) (π (h), π (h) ) s i j h, ε h = 0, 1, for h = 1,..., n; III. blocks of the form à B for any pair à π(h) and i B π (n+h) with ε j h d (h) (π (h), π (h) ) s i j h, ε h = 0, 1, for h = 1,..., n. Denote z h := (2s h + 1 ε h ) for h = 1,..., n.

14 Verification: CASE k i 3 The blocks containing points a, b, c X (resp. ã, b, c X): Λ + n z h λ (h) b (n+h) + z h λ (n+h) b (h) h=1 The blocks containing points a, b, c with a, b X and c, X (resp. ã, b, c): n z h λ (h) 2 u n+h + z h λ (n+h) 2 u h h=1 The defined blocks will form a 3-design if Λ + n z h λ (h) b (n+h) + z h λ (n+h) b (h) = h=1 equivalently n h=1 z h λ (h) 2 u n+h + z h λ (n+h) 2 u h, n Λ = {(λ (h) 2 u h ) (λ (h) b (n+h) + λ (n+h) b (h) )}z h. h=1

15 Verification: CASE K 1 = 2 The condition for which the defined blocks form a 3-designs becomes Λ = {a 1 u n+1 + λ (n+1) 2 u 1 λ (n+1) b (1) }z 1 n + {(λ (h) 2 u n+h + λ(n+h) 2 u h ) (λ (h) b (n+h) + λ (n+h) b (h) )}z h. h=2

16 Summary of Construction I (i) If k 1 = 2 and 0 = {a 1 u n+1 + λ (n+1) 2 u 1 λ (n+1) b (1) }z 1 n + {(λ (h) 2 u h ) (λ (h) b (n+h) + λ (n+h) b (h) )}z h,(1) h=2 with 1 z h w h if both D h and D n+h are simple and 1 z h t h if D h or D n+h is non-simple, then there exists a 3 (2v, k, Θ) design with n Θ = {a 1 u n+1 + λ (n+1) 2 u 1 }z 1 + {(λ (h) 2 u h )}z h. h=2 (ii) If k 1 3 and n 0 = {(λ (h) 2 u h ) (λ (h) b (n+h) + λ (n+h) b (h) )}z h, (2) h=1 with 1 z h w h if both D h and D n+h are simple and 1 z h t h if D h or D n+h is non-simple, then there exists a 3 (2v, k, Θ) design with n Θ = {(λ (h) 2 u h )}z h. h=1

17 Summary of Construction I (Cont.) (iii) If k 1 = 2 and 0 < {a 1 u n+1 + λ (n+1) 2 u 1 λ (n+1) b (1) }z 1 n + {(λ (h) 2 u h ) (λ (h) b (n+h) + λ (n+h) b (h) )}z h,(3) h=2 with 1 z h w h if both D h and D n+h are simple and 1 z h t h if D h or D n+h is non-simple, further if there is a 3 (v, k, Λ) design having Λ = {a 1 u n+1 + λ (n+1) 2 u 1 λ (n+1) b (1) }z 1 n + {(λ (h) 2 u h ) (λ (h) b (n+h) + λ (n+h) b (h) )}z h,(4) h=2 then there exists a 3 (2v, k, Θ) design with n Θ = {a 1 u n+1 + λ (n+1) 2 u 1 }z 1 + {(λ (h) 2 u h )}z h. h=2

18 Summary of Construction I (Cont.) (iv) If k 1 3 and 0 < n {(λ (h) 2 u h ) (λ (h) b (n+h) + λ (n+h) b (h) )}z h, (5) h=1 with 1 z h w h if both D h and D n+h are simple and 1 z h t h if D h or D n+h is non-simple, further if there is a 3 (v, k, Λ) design having Λ = n {(λ (h) 2 u h ) (λ (h) b (n+h) + λ (n+h) b (h) )}z h, (6) h=1 then there exists a 3 (2v, k, Θ) design with n Θ = {(λ (h) 2 u h )}z h. h=1

19 Construction II Construction II deals with the case k n = k/2. Take D n = D 2n. Blocks of types I, II, and III are as in Construction I for h = 1,..., n 1. Define a further type of blocks. IV. blocks of the form A B for any pair A π (n) i with ε n d (n) (π (n), π (n) ) s i j n, ε n = 0, 1. and B π (2n) j

20 Summary of Construction II (i) If k 1 = 2 and 0 = (a 1 u n+1 + λ (n+1) 2 u 1 λ (n+1) b (1) )z 1 + (λ (n) 2 u n λ (n) b (n) )z n n 1 + {(λ (h) 2 u h ) (λ (h) b (n+h) + λ (n+h) b (h) )}z h,(7) h=2 with 1 z h w h if both D h and D n+h are simple and 1 z h t h if D h or D n+h is non-simple, then there exists a 3 (2v, k, Θ) design with Θ = (a 1 u n+1 + λ (n+1) 2 u 1 )z 1 + (λ (n) n 1 2 u n)z n + {(λ (h) 2 u h )}z h. h=2 (ii) If k 1 3 and 0 = (λ (n) 2 u n λ (n) b (n) )z n n 1 + {(λ (h) 2 u h ) (λ (h) b (n+h) + λ (n+h) b (h) )}z h,(8) h=1 with 1 z h w h if both D h and D n+h are simple and 1 z h t h if D h or D n+h is non-simple, then there exists a 3 (2v, k, Θ) design with Θ = (λ (n) n 1 2 u n)z n + {(λ (h) 2 u h )}z h. h=1

21 Summary of Construction II (Cont.) (iii) If k 1 = 2 and 0 < (a 1 u n+1 + λ (n+1) 2 u 1 λ (n+1) b (1) )z 1 + (λ (n) 2 u n λ (n) b (n) )z n n 1 + {(λ (h) 2 u h ) (λ (h) b (n+h) + λ (n+h) b (h) )}z h,(9) h=2 with 1 z h w h if both D h and D n+h are simple and 1 z h t h if D h or D n+h is non-simple, further if there is a 3 (v, k, Λ) design having Λ = (a 1 u n+1 + λ (n+1) 2 u 1 λ (n+1) b (1) )z 1 + (λ (n) 2 u n λ (n) b (n) )z n n 1 + {(λ (h) 2 u h ) (λ (h) b (n+h) + λ (n+h) b (h) )}z h (10), h=2 then there exists a 3 (2v, k, Θ) design with Θ = (a 1 u n+1 + λ (n+1) 2 u 1 )z 1 + (λ (n) n 1 2 u n)z n + {(λ (h) 2 u h )}z h. h=2

22 Summary of Construction II (Cont.) (iv) If k 1 3 and 0 < (λ (n) 2 u n λ (n) b (n) )z n n 1 + {(λ (h) 2 u h ) (λ (h) b (n+h) + λ (n+h) b (h) )}z h (11), h=1 with 1 z h w h if both D h and D n+h are simple and 1 z h t h if D h or D n+h is non-simple, further if there is a 3 (v, k, Λ) design having Λ = (λ (n) 2 u n λ (n) b (n) )z n n 1 + {(λ (h) 2 u h ) (λ (h) b (n+h) + λ (n+h) b (h) )}z h (12), h=1 then there exists a 3 (2v, k, Θ) design with Θ = (λ (n) n 1 2 u n)z n + {(λ (h) 2 u h )}z h. h=1

23 APPLICATIONS For applications of Constructions I and II we implicitly use the following result and observation. Baranyai-Theorem The trivial k (v, k, 1) design is (1,1)-resolvable (i.e. having a parallelism) if and only if k v. Block orbits If gcd(v, k) = 1, then the k (v, k, 1) design is (1, v)-resolvable. (The resolvable classes are the block orbits of a fixed point free automorphism of order v.)

24 APPLICATIONS F1 Construction II with n = 1. v, k : integers with v > k 3 and gcd(v, k) = 1. D 1 : the complete design 3 (v, k, ( v 3) k 3 ). Then λ (1) = ( v 3) (1) k 3, λ 2 = ( v 2 k 2), u1 = k, b (1) = v, and w 1 = ( v 1 k 1) /k. D: 3 (v, 2k, Λ). Construction II yields a simple 3-design 3 (2v, 2k, Θ) when it holds or (λ (1) 2 u 1 λ (1) b (1) )z 1 = Λ, with z 1 ( v 1 k 1) /k. Then ( ) v 3 z 1 = Λ/2 k 2 is an integer, Θ = λ (1) 2 u k(v 2)Λ 1z 1 = 2(v k).

25 APPLICATIONS F1 (Cont.) Take the complete design D: 3 (v, 2k, Λ) := 3 (v, 2k, ( v 3 2k 3) ). If ( ) ( ) v 3 v 3 z 1 = /2 2k 3 k 2 is an integer, with z 1 ( v 1 k 1) /k. Then there is a simple 3 (2v, 2k, Θ) design with ( ) k(v 2) v 3 Θ =. 2(v k) 2k 3

26 APPLICATIONS F1 Some special cases: k = 3, 4, 5. 1 There exists a simple 3 (2v, 6, Θ) design with ( ) 3(v 2) v 3 Θ =, 2(v 3) 3 for v 1, 4, 5, 8 mod There exists a simple 3 (2v, 8, Θ) design with ( ) 4(v 2) v 3 Θ =, 2(v 4) 5 for v 1, 5, 7, 11, 15, 17 mod There exists a simple 3 (2v, 10, Θ) design with ( ) 5(v 2) v 3 Θ =, 2(v 5) 7 for v 0, 1, 2, 6 mod 7, and v 0, 1, 6, 7 mod 8, and gcd(v, 5) = 1.

27 APPLICATIONS F2 Construction II with n = 2, k 1 = 2. v, k : integers with v > 2k, k 3, gcd(v, 2k) = 1 & gcd(v, k + 1) = 1. C 1 : 2 (v, 2, 1); α (1) = 0, α (1) 2 = 1, u 1 = 2, b (1) = v, t 1 = (v 1)/2, ( 1 a 1 = v 2 k(2k 1) 2k 2). D1 is a union of a 1 copies of C 1. D 3 : 2 (v, 2k, ( v 3 ) 2k 3 ); λ (3) = ( v 3 ) (3) 2k 3, λ 2 = ( v 2 2k 2), u3 = 2k, b (3) = v, w 3 = 1 ( v 1 2k 2k 1). D 2 : 2 (v, k + 1, ( v 3) k 2 ); λ (2) = ( v 3) (2) k 2, λ 2 = ( v 2 k 1), u2 = k + 1, b (2) = v, w 2 = 1 ( v 1 ) k+1 k.

28 APPLICATIONS F2 (Cont.) Set A := A 1 z 1 + A 2 z 2, where A 1 = (a 1 u 3 + λ (3) 2 u 1 λ (3) b (1) ), A 2 = (λ (2) 2 u 2 λ (2) b (2) ). Then ( ) v 3 v(4k 2 10k + 2) + 8k A 1 =, 2k 3 (2k 1)(2k 2) ( ) v 3 (v k 1) A 2 = 2. k 2 (k 1) For any integer z 1 with 1 z 1 w 1 we have A = 0 iff z 2 = A 1 z 1 /A 2 If z 2 is an integer with z 2 w 2,then there is a simple 3 (2v, 2(k + 1), Θ) design with ( ( ) ( )) ( ) 2k v 2 v 2 v 2 Θ = + 2 z k(2k 1) 2k 2 2k z k 1 2.

29 APPLICATIONS F2 (Cont.) An example: z 1 = 1. Then ( ) v k 2 z 2 = k k! 2.k(k + 1)... (2k 3) v(4k 2 10k + 2) + 8k. (2k 1)(2k 2) If z 2 is an integer and z 2 w 2, then there is a simple 3 (2v, 2(k + 1), Θ) design with ( ) ( ) 4k v 2 v 2 Θ = + (k + 1).z (2k 1) 2k 2 k 1 2.

30 APPLICATIONS F2 (Cont.) Two special cases: k = 3, 4 with z 1 = 1. 1 There exists a simple 3 (2v, 8, Θ) design with for all v 5, 17, 35, 47 mod 60. Θ = 7 v(v 2)(v 3)(v 5), 30 2 There exists a simple 3 (2v, 10, Θ) design with ( ) v 2 Θ = 81v /7(v 5), 6 for all v 7, 23, 63, 111, 167, 191, 223, 231, 247 mod 280.

31 APPLICATIONS F3 Some more examples 1 There exists a simple 3 (2v, 5, 3 4 (v 2)(v 3)) design when v 2 mod 6. 2 There exists a simple 3 (2v, 7, 48 5 ( v 2 ) 3 (11v 52)) design for all v 4, 76, 112, 148 mod There exists a simple 3 (2(2 f + 1), 5, 15(2 f 1) design for f odd. 4 There exists a simple 3 (2(2 f + 1), 6, (2 f 1).m) design with m = 5, 30, 35, 45, 50, 75, 80 and gcd(f, 6) = 1.

32 (1, σ)-resolvability For each pair (D i, D n+i ) define σ (i) = u i b (n+i) + u n+i b (i). For the pair (D n, D n ) in Construction II define σ (n) = u n b (n). Let m 1,..., m n be integers such that m i σ (i) = m j σ (j) := σ for i, j = 1,..., n. If a 3 (v, 2k, Λ) design D is required in the construction, it is assumed that D is (1, σ)-resolvable. Assume that the blocks constructed by using each pair (D i, D n+i ) can be partitioned into 1 (v, 2k, σ) designs. Then the designs obtained from Constructions I and II are (1, σ)-resolvable.

33 (1, σ)-resolvability Some examples The 3 (2v, 6, 1 4 (v 2)(v 4)(v 5)) designs in F1 are (1, 3v)-resolvable when v 1, 4, 5, 13, 20, 28, 29, 32 mod 36. The 3 (2v, 8, Θ) designs with Θ = 30 7 v(v 2)(v 3)(v 5), and v 5, 17, 35, 47 mod 60 in F2 are (1, 8v)-resolvable. The 3 (2v, 10, Θ) designs with Θ = 81v ( v 2) 6 /7(v 5), and v 7, 23, 63, 111, 167, 191, 223, 231, 247 mod 280 in F2 are (1, 10v)-resolvable, when 16 (v 7). The 3 (2v, 5, 3 4 (v 2)(v 4)) designs in F3 are (1, 5v)-resolvable, when v 2, 26, 104, 128 mod 150.

34 References Zs. Baranyai. On the factorization of the complete uniform hypergraph. In Proc. Erdös Colloquium, Keszthely, 1973, North-Holland, Amstersdam (1975), J. Bierbrauer. Some friends of Alltop s designs 4 (2 f + 1, 5, 5). J. Combin. Math. Combin. Comput. 36 (2001) J. Bierbrauer, TvT. Shadow and shade of designs 4 (2 f + 1, 6, 10) (1992) unpublished manuscript. M. Jimbo, Y. Kunihara, R. Laue, M. Sawa. Unifying some known infinite families of combinatorial 3-designs. J. Combin. Theory, Ser.A 118 (2011) R. Laue. Resolvable t-designs. Des. Codes Cryptogr. 32 (2004) TvT. Recursive constructions for 3-designs and resolvable 3-designs, J. Statist. Plann. Inference 95 (2001), TvT. Construction of 3-designs using parallelism, J. Geom. 67 (2000)