APPLICATIONS For applications of Constructions I and II we - PowerPoint PPT Presentation

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

O UTLINE � Generic constructions � Applications � ( 1 , σ ) -resolvable 3-designs

G ENERIC CONSTRUCTIONS 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

G ENERIC CONSTRUCTIONS 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 |} .

G ENERIC CONSTRUCTIONS • n � 1, integer. • { k 1 , . . . , k n , k n + 1 , . . . , k 2 n } and k , integers, such that 2 � k 1 < . . . < k n < k / 2 and k i + k n + i = k for i = 1 , . . . , n .

G ENERIC CONSTRUCTIONS • n � 1, integer. • { k 1 , . . . , k n , k n + 1 , . . . , k 2 n } 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 2 n 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 .

G ENERIC CONSTRUCTIONS • n � 1, integer. • { k 1 , . . . , k n , k n + 1 , . . . , k 2 n } 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 2 n 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.

G ENERIC CONSTRUCTIONS • n � 1, integer. • { k 1 , . . . , k n , k n + 1 , . . . , k 2 n } 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 2 n 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 ) .

G ENERIC CONSTRUCTIONS • 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 ) π ( j ) 1 , . . . , π ( j ) π ( j ) 1 , . . . , π ( j ) t j , t j , . . . , t j

G ENERIC CONSTRUCTIONS • 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 ) π ( j ) 1 , . . . , π ( j ) π ( j ) 1 , . . . , π ( j ) t j , t 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.

G ENERIC CONSTRUCTIONS • 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 ) π ( j ) 1 , . . . , π ( j ) π ( j ) 1 , . . . , π ( j ) t j , t 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 ) .

G ENERIC CONSTRUCTIONS Notation: • π ( ℓ ) 1 , . . . , π ( ℓ ) w ℓ : the w ℓ classes in a ( 1 , σ ( ℓ ) ) -resolution of D ℓ , ℓ = 1 , . . . , 2 n . 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 ℓ − | i − j |} . 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.

G ENERIC CONSTRUCTIONS Construction I Let ˜ D i = ( ˜ B i ) be a copy of D i defined on ˜ X , ˜ 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 ; B for any pair A ∈ π ( h ) II. blocks of the form A ∪ ˜ and i π ( n + h ) with ε h � d ( h ) ( π ( h ) , π ( h ) ˜ B ∈ ˜ ) � s h , ε h = 0 , 1, for j i j h = 1 , . . . , n ; π ( h ) III. blocks of the form ˜ A ∪ B for any pair ˜ A ∈ ˜ and i B ∈ π ( n + h ) with ε h � d ( h ) ( π ( h ) , π ( h ) ) � s h , ε h = 0 , 1, for j i j h = 1 , . . . , n . Denote z h := ( 2 s h + 1 − ε h ) for h = 1 , . . . , n .

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

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

G ENERIC CONSTRUCTIONS Summary of Construction I (i) If k 1 = 2 and { a 1 u n + 1 + λ ( n + 1 ) u 1 − λ ( n + 1 ) b ( 1 ) } z 1 0 = 2 n { ( λ ( h ) 2 u n + h + λ ( n + h ) u h ) − ( λ ( h ) b ( n + h ) + λ ( n + h ) b ( h ) ) } z h , (1) � + 2 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 − ( 2 v , k , Θ ) design with n Θ = { a 1 u n + 1 + λ ( n + 1 ) { ( λ ( h ) 2 u n + h + λ ( n + h ) � u 1 } z 1 + u h ) } z h . 2 2 h = 2 (ii) If k 1 � 3 and n u h ) − ( λ ( h ) b ( n + h ) + λ ( n + h ) b ( h ) ) } z h , (2) { ( λ ( h ) 2 u n + h + λ ( n + h ) � 0 = 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 − ( 2 v , k , Θ ) design with n { ( λ ( h ) 2 u n + h + λ ( n + h ) � Θ = u h ) } z h . 2 h = 1

G ENERIC CONSTRUCTIONS Summary of Construction I (Cont.) (iii) If k 1 = 2 and { a 1 u n + 1 + λ ( n + 1 ) u 1 − λ ( n + 1 ) b ( 1 ) } z 1 0 < 2 n { ( λ ( h ) 2 u n + h + λ ( n + h ) u h ) − ( λ ( h ) b ( n + h ) + λ ( n + h ) b ( h ) ) } z h , (3) � + 2 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 { a 1 u n + 1 + λ ( n + 1 ) u 1 − λ ( n + 1 ) b ( 1 ) } z 1 Λ = 2 n u h ) − ( λ ( h ) b ( n + h ) + λ ( n + h ) b ( h ) ) } z h , { ( λ ( h ) 2 u n + h + λ ( n + h ) � (4) + 2 h = 2 then there exists a 3 − ( 2 v , k , Θ ) design with n Θ = { a 1 u n + 1 + λ ( n + 1 ) { ( λ ( h ) 2 u n + h + λ ( n + h ) � u 1 } z 1 + u h ) } z h . 2 2 h = 2