6.7 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Block design | 4/6 | https://en.wikipedia.org/wiki/Block_design | reference | science, encyclopedia | 2026-05-05T09:49:03.148970+00:00 | kb-cron |
== General balanced designs (t-designs) == 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
λ
i
=
λ
(
v
−
i
t
−
i
)
/
(
k
−
i
t
−
i
)
for
i
=
0
,
1
,
…
,
t
,
{\displaystyle \lambda _{i}=\lambda \left.{\binom {v-i}{t-i}}\right/{\binom {k-i}{t-i}}{\text{ for }}i=0,1,\ldots ,t,}
where λi is the number of blocks that contain any i-element set of points and λt = λ. Note that
b
=
λ
0
=
λ
(
v
t
)
/
(
k
t
)
{\displaystyle b=\lambda _{0}=\lambda {v \choose t}/{k \choose t}}
and
r
=
λ
1
=
λ
(
v
−
1
t
−
1
)
/
(
k
−
1
t
−
1
)
{\displaystyle r=\lambda _{1}=\lambda {v-1 \choose t-1}/{k-1 \choose t-1}}
. 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.) A consequence of this theorem is that every t-design with t ≥ 2 is also a 2-design. A t-(v,k,1)-design is called a Steiner system. The term block design by itself usually means a 2-design.
=== Derived and extendable t-designs === 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. Theorem: If a t-(v,k,λ) design has an extension, then k + 1 divides b(v + 1). The only extendable projective planes (symmetric 2-(n2 + n + 1, n + 1, 1) designs) are those of orders 2 and 4. Every Hadamard 2-design is extendable (to an Hadamard 3-design). Theorem:. If D, a symmetric 2-(v,k,λ) design, is extendable, then one of the following holds:
D is an Hadamard 2-design, v = (λ + 2)(λ2 + 4λ + 2), k = λ2 + 3λ + 1, v = 495, k = 39, λ = 3. 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.
==== Inversive planes ==== 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. 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. An example of an ovoid is the elliptic quadric, the set of zeros of the quadratic form
x1x2 + f(x3, x4), where f is an irreducible quadratic form in two variables over GF(q). [f(x,y) = x2 + xy + y2 for example]. If q is an odd power of 2, another type of ovoid is known – the Suzuki–Tits ovoid. 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).
== Partially balanced designs (PBIBDs) == 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: