ABSTRACT
A collection of subsets B https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429280092/903cf4ed-cbd9-4ecc-bd90-95916cd1752b/content/inline-math17_1.jpg"/> called blocks chosen from a set X of v points is said to be t-wise balanced if every t-element subset of X is contained in precisely λ blocks, for some positive integer λ called the index. Designs with index 1 are called Steiner designs and are the most desirable. A t-wise balanced structure ( X , B ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429280092/903cf4ed-cbd9-4ecc-bd90-95916cd1752b/content/inline-math17_2.jpg"/> of points and blocks is denoted with parameters t- ( v , K , λ ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429280092/903cf4ed-cbd9-4ecc-bd90-95916cd1752b/content/inline-math17_3.jpg"/> , where K https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429280092/903cf4ed-cbd9-4ecc-bd90-95916cd1752b/content/inline-math17_4.jpg"/> contains the sizes of the blocks that were used. If K = { k } https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429280092/903cf4ed-cbd9-4ecc-bd90-95916cd1752b/content/inline-math17_5.jpg"/> , then a t-wise balanced design is simply called a t-design with parameters t-(v, k, λ) or classically as S λ (t, k, v). For example: X = { 1 , 2 , 3 , … , 10 , 11 } B = { { 1 , 6 , 7 } , { 1 , 8 , 11 } , { 1 , 9 , 10 } , { 2 , 6 , 8 } , { 2 , 7 , 9 } , { 2 , 10 , 11 } , { 3 , 6 , 9 } , { 3 , 7 , 11 } , { 3 , 8 , 10 } , { 4 , 6 , 10 } , { 4 , 7 , 8 } , { 4 , 9 , 11 } , { 5 , 6 , 11 } , { 5 , 7 , 10 } , { 5 , 8 , 9 } , { 1 , 2 , 3 , 4 , 5 } } https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429280092/903cf4ed-cbd9-4ecc-bd90-95916cd1752b/content/umath17_1.jpg"/> is a 2-(11, {3, 5}, 1) design. A t-wise balanced design is said to be a proper design if ( X k ) ⊈ B https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429280092/903cf4ed-cbd9-4ecc-bd90-95916cd1752b/content/inline-math17_6.jpg"/> , for any k ∈ K https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429280092/903cf4ed-cbd9-4ecc-bd90-95916cd1752b/content/inline-math17_7.jpg"/> . (If ( X k ) ⊆ B https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429280092/903cf4ed-cbd9-4ecc-bd90-95916cd1752b/content/inline-math17_8.jpg"/> , then B ∖ ( X k ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429280092/903cf4ed-cbd9-4ecc-bd90-95916cd1752b/content/inline-math17_9.jpg"/> remains t-wise balanced.) In 302particular the point set X is excluded from being a block. A t-wise balanced design of index λ > 1 can always be constructed from a design with index 1 by repeating all the blocks λ times. A t-wise balanced design with no repeated blocks is said to be a simple design. We will consider only simple proper designs in this chapter.
