In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as blocks, chosen such that frequency of the elements[clarification needed] satisfies certain conditions making the collection of blocks exhibit symmetry (balance).
Block designs have applications in many areas, including experimental design, finite geometry, physical chemistry, software testing, cryptography, and algebraic geometry.
A design is said to be balanced (up to t) if all t-subsets of the original set occur in equally many (i.e., λ) blocks[clarification needed].
When t is unspecified, it can usually be assumed to be 2, which means that each pair of elements is found in the same number of blocks and the design is pairwise balanced.
The incidence matrix of a non-binary design lists the number of times each element is repeated in each block.
The simplest type of "balanced" design (t=1) is known as a tactical configuration or 1-design.
(To avoid degenerate examples, it is also assumed that v > k, so that no block contains all the elements of the set.
A rather surprising and not very obvious (but very general) combinatorial result for these designs is that if points are denoted by any arbitrarily chosen set of equally or unequally spaced numerics, there is no choice of such a set which can make all block-sums (that is, sum of all points in a given block) constant.
The unique (6,3,2)-design (v = 6, k = 3, λ = 2) has 10 blocks (b = 10) and each element is repeated 5 times (r = 5).
[8] Using the symbols 0 − 5, the blocks are the following triples: and the corresponding incidence matrix (a v×b binary matrix with constant row sum r and constant column sum k) is: One of four nonisomorphic (8,4,3)-designs has 14 blocks with each element repeated 7 times.
Its corresponding incidence matrix can also be symmetric, if the labels or blocks are sorted the right way: The case of equality in Fisher's inequality, that is, a 2-design with an equal number of points and blocks, is called a symmetric design.
[11] The parameters of a symmetric design satisfy This imposes strong restrictions on v, so the number of points is far from arbitrary.
The Bruck–Ryser–Chowla theorem gives necessary, but not sufficient, conditions for the existence of a symmetric design in terms of these parameters.
They form the only known infinite family (with respect to having a constant λ value) of symmetric block designs.
Given an Hadamard matrix of size 4a in standardized form, remove the first row and first column and convert every −1 to a 0.
The set of parallel classes is called a resolution of the design.
[20] Consequently, a symmetric design can not have a non-trivial (more than one parallel class) resolution.
A solution of the famous 15 schoolgirl problem is a resolution of a 2-(15,3,1) design.
The equations are where λi is the number of blocks that contain any i-element set of points and λt = λ.
[27] If D, a symmetric 2-(v,k,λ) design, is extendable, then one of the following holds: 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.
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.
(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.
(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).
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.
[33] They fall into six types[34] based on a classification of the then known PBIBD(2)s by Bose & Shimamoto (1952):[35] The mathematical subject of block designs originated in the statistical framework of design of experiments.
The rows of their incidence matrices are also used as the symbols in a form of pulse-position modulation.
They coat two different sunscreens on the upper sides of the hands of a test person.
After a UV radiation they record the skin irritation in terms of sunburn.
The number of treatments is 3 (sunscreens) and the block size is 2 (hands per person).
It is impossible to use a complete design (all treatments in each block) in this example because there are 3 sunscreens to test, but only 2 hands on each person.