236 lines
6.7 KiB
Markdown
236 lines
6.7 KiB
Markdown
---
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title: "Block design"
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chunk: 4/6
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source: "https://en.wikipedia.org/wiki/Block_design"
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category: "reference"
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tags: "science, encyclopedia"
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date_saved: "2026-05-05T09:49:03.148970+00:00"
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instance: "kb-cron"
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---
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== General balanced designs (t-designs) ==
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Given any positive integer t, a t-design B is a class of k-element subsets of X, called blocks, such that every point x in X appears in exactly r blocks, and every t-element subset T appears in exactly λ blocks. The numbers v (the number of elements of X), b (the number of blocks), k, r, λ, and t are the parameters of the design. The design may be called a t-(v,k,λ)-design. Again, these four numbers determine b and r and the four numbers themselves cannot be chosen arbitrarily. The equations are
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λ
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λ
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)
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for
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0
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,
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1
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,
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…
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,
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t
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,
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{\displaystyle \lambda _{i}=\lambda \left.{\binom {v-i}{t-i}}\right/{\binom {k-i}{t-i}}{\text{ for }}i=0,1,\ldots ,t,}
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where λi is the number of blocks that contain any i-element set of points and λt = λ.
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Note that
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b
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λ
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0
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λ
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{\displaystyle b=\lambda _{0}=\lambda {v \choose t}/{k \choose t}}
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and
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r
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λ
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1
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{\displaystyle r=\lambda _{1}=\lambda {v-1 \choose t-1}/{k-1 \choose t-1}}
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.
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Theorem: Any t-(v,k,λ)-design is also an s-(v,k,λs)-design for any s with 1 ≤ s ≤ t. (Note that the "lambda value" changes as above and depends on s.)
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A consequence of this theorem is that every t-design with t ≥ 2 is also a 2-design.
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A t-(v,k,1)-design is called a Steiner system.
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The term block design by itself usually means a 2-design.
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=== Derived and extendable t-designs ===
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Let D = (X, B) be a t-(v,k,λ) design and p a point of X. The derived design Dp has point set X − {p} and as block set all the blocks of D which contain p with p removed. It is a (t − 1)-(v − 1, k − 1, λ) design. Note that derived designs with respect to different points may not be isomorphic. A design E is called an extension of D if E has a point p such that Ep is isomorphic to D; we call D extendable if it has an extension.
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Theorem: If a t-(v,k,λ) design has an extension, then k + 1 divides b(v + 1).
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The only extendable projective planes (symmetric 2-(n2 + n + 1, n + 1, 1) designs) are those of orders 2 and 4.
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Every Hadamard 2-design is extendable (to an Hadamard 3-design).
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Theorem:.
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If D, a symmetric 2-(v,k,λ) design, is extendable, then one of the following holds:
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D is an Hadamard 2-design,
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v = (λ + 2)(λ2 + 4λ + 2), k = λ2 + 3λ + 1,
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v = 495, k = 39, λ = 3.
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Note that the projective plane of order two is an Hadamard 2-design; the projective plane of order four has parameters which fall in case 2; the only other known symmetric 2-designs with parameters in case 2 are the order 9 biplanes, but none of them are extendable; and there is no known symmetric 2-design with the parameters of case 3.
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==== Inversive planes ====
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A design with the parameters of the extension of an affine plane, i.e., a 3-(n2 + 1, n + 1, 1) design, is called a finite inversive plane, or Möbius plane, of order n.
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It is possible to give a geometric description of some inversive planes, indeed, of all known inversive planes. An ovoid in PG(3,q) is a set of q2 + 1 points, no three collinear. It can be shown that every plane (which is a hyperplane since the geometric dimension is 3) of PG(3,q) meets an ovoid O in either 1 or q + 1 points. The plane sections of size q + 1 of O are the blocks of an inversive plane of order q. Any inversive plane arising this way is called egglike. All known inversive planes are egglike.
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An example of an ovoid is the elliptic quadric, the set of zeros of the quadratic form
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x1x2 + f(x3, x4),
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where f is an irreducible quadratic form in two variables over GF(q). [f(x,y) = x2 + xy + y2 for example].
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If q is an odd power of 2, another type of ovoid is known – the Suzuki–Tits ovoid.
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Theorem. Let q be a positive integer, at least 2. (a) If q is odd, then any ovoid is projectively equivalent to the elliptic quadric in a projective geometry PG(3,q); so q is a prime power and there is a unique egglike inversive plane of order q. (But it is unknown if non-egglike ones exist.) (b) if q is even, then q is a power of 2 and any inversive plane of order q is egglike (but there may be some unknown ovoids).
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== Partially balanced designs (PBIBDs) ==
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An n-class association scheme consists of a set X of size v together with a partition S of X × X into n + 1 binary relations, R0, R1, ..., Rn. A pair of elements in relation Ri are said to be ith–associates. Each element of X has ni ith associates. Furthermore: |