Block design

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).[5] Using the symbols 0 − 5, the blocks are the following triples:

012    013    024    035    045    125    134    145    234    235.

and the corresponding incidence matrix (a v×b binary matrix with constant row sum r and constant column sum k) is:

${\displaystyle {\begin{pmatrix}1&1&1&1&1&0&0&0&0&0\\1&1&0&0&0&1&1&1&0&0\\1&0&1&0&0&1&0&0&1&1\\0&1&0&1&0&0&1&0&1&1\\0&0&1&0&1&0&1&1&1&0\\0&0&0&1&1&1&0&1&0&1\\\end{pmatrix}}}$

One of four nonisomorphic (8,4,3)-designs has 14 blocks with each element repeated 7 times. Using the symbols 0 − 7 the blocks are the following 4-tuples:[5]

0123    0124    0156    0257    0345    0367    0467    1267    1346    1357    1457    2347    2356    2456.

The unique (7,3,1)-design is symmetric and has 7 blocks with each element repeated 3 times. Using the symbols 0 − 6, the blocks are the following triples:[5]

013    026    045    124    156    235    346.

This design is associated with the Fano plane, with the elements and blocks of the design corresponding to the points and lines of the plane. Its corresponding incidence matrix can also be symmetric, if the labels or blocks are sorted the right way:

${\displaystyle \left({\begin{matrix}1&1&1&0&0&0&0\\1&0&0&1&1&0&0\\1&0&0&0&0&1&1\\0&1&0&1&0&1&0\\0&1&0&0&1&0&1\\0&0&1&1&0&0&1\\0&0&1&0&1&1&0\end{matrix}}\right)}$

Finite projective planes are symmetric 2-designs with λ = 1 and order n > 1. For these designs the symmetric design equation becomes:

${\displaystyle v-1=k(k-1).}$

Since k = r we can write the order of a projective plane as n = k − 1 and, from the displayed equation above, we obtain v = (n + 1)n  +  1 = n²  +  n  +  1 points in a projective plane of order n.

As a projective plane is a symmetric design, we have b = v, meaning that b = n²  +  n  +  1 also. The number b is the number of lines of the projective plane. There can be no repeated lines since λ = 1, so a projective plane is a simple 2-design in which the number of lines and the number of points are always the same. For a projective plane, k is the number of points on each line and it is equal to n + 1. Similarly, r = n + 1 is the number of lines with which a given point is incident.

For n = 2 we get a projective plane of order 2, also called the Fano plane, with v = 4 + 2 + 1 = 7 points and 7 lines. In the Fano plane, each line has n + 1 = 3 points and each point belongs to n + 1 = 3 lines.

Projective planes are known to exist for all orders which are prime numbers or powers of primes. They form the only known infinite family (with respect to having a constant λ value) of symmetric block designs.[9]

A biplane or biplane geometry is a symmetric 2-design with λ = 2; that is, every set of two points is contained in two blocks ("lines"), while any two lines intersect in two points.[9] They are similar to finite projective planes, except that rather than two points determining one line (and two lines determining one point), two points determine two lines (respectively, points). A biplane of order n is one whose blocks have k = n + 2 points; it has v = 1 + (n + 2)(n + 1)/2 points (since r = k).

The 18 known examples[10] are listed below.

• (Trivial) The order 0 biplane has 2 points (and lines of size 2; a 2-(2,2,2) design); it is two points, with two blocks, each consisting of both points. Geometrically, it is the digon.
• The order 1 biplane has 4 points (and lines of size 3; a 2-(4,3,2) design); it is the complete design with v = 4 and k = 3. Geometrically, the points are the vertices of a tetrahedron and the blocks are its faces.
• The order 2 biplane is the complement of the Fano plane: it has 7 points (and lines of size 4; a 2-(7,4,2)), where the lines are given as the complements of the (3-point) lines in the Fano plane.[11]
• The order 3 biplane has 11 points (and lines of size 5; a 2-(11,5,2)), and is also known as the Paley biplane after Raymond Paley; it is associated to the Paley digraph of order 11, which is constructed using the field with 11 elements, and is the Hadamard 2-design associated to the size 12 Hadamard matrix; see Paley construction I.

Algebraically this corresponds to the exceptional embedding of the projective special linear group PSL(2,5) in PSL(2,11) – see projective linear group: action on p points for details.[12]

• There are three biplanes of order 4 (and 16 points, lines of size 6; a 2-(16,6,2)). One is the Kummer configuration. These three designs are also Menon designs.
• There are four biplanes of order 7 (and 37 points, lines of size 9; a 2-(37,9,2)).[13]
• There are five biplanes of order 9 (and 56 points, lines of size 11; a 2-(56,11,2)).[14]
• Two biplanes are known of order 11 (and 79 points, lines of size 13; a 2-(79,13,2)).[15]

Biplanes of orders 5, 6, 8 and 10 do not exist, as shown by the Bruck-Ryser-Chowla theorem.

An Hadamard matrix of size m is an m × m matrix H whose entries are ±1 such that HH^⊤  = mI_m, where H^⊤ is the transpose of H and I_m is the m × m identity matrix. An Hadamard matrix can be put into standardized form (that is, converted to an equivalent Hadamard matrix) where the first row and first column entries are all +1. If the size m > 2 then m must be a multiple of 4.

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 resulting 0–1 matrix M is the incidence matrix of a symmetric 2-(4a − 1, 2a − 1, a − 1) design called an Hadamard 2-design.[16] It contains ${\displaystyle 4a-1}$ blocks/points; each contains/is contained in ${\displaystyle 2a-1}$ points/blocks. Each pair of points is contained in exactly ${\displaystyle a-1}$ blocks.

This construction is reversible, and the incidence matrix of a symmetric 2-design with these parameters can be used to form an Hadamard matrix of size 4a.

Let D = (X, B) be a t-(v,k,λ) design and p a point of X. The derived design D_p 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 E_p is isomorphic to D; we call D extendable if it has an extension.

Theorem:[21] 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.[22]

Every Hadamard 2-design is extendable (to an Hadamard 3-design).[23]

Theorem:.[24] If D, a symmetric 2-(v,k,λ) design, is extendable, then one of the following holds:

1. D is an Hadamard 2-design,
2. v  =  (λ + 2)(λ² + 4λ + 2), k = λ² + 3λ + 1,
3. 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.[25]

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).

Let A(3) be the following association scheme with three associate classes on the set X = {1,2,3,4,5,6}. The (i,j) entry is s if elements i and j are in relation R_s.

The blocks of a PBIBD(3) based on A(3) are:

The parameters of this PBIBD(3) are: v  =  6, b  =  8, k  =  3, r  =  4 and λ₁ = λ₂ = 2 and λ₃ = 1. Also, for the association scheme we have n₀  =  n₂  =  1 and n₁  =  n₃  =  2.[27] The incidence matrix M is

${\displaystyle {\begin{pmatrix}1&1&1&1&0&0&0&0\\1&1&0&0&1&1&0&0\\0&0&1&1&1&1&0&0\\1&0&1&0&0&0&1&1\\0&1&0&0&1&0&1&1\\0&0&0&1&0&1&1&1\\\end{pmatrix}}}$

and the concurrence matrix MM^T is

${\displaystyle {\begin{pmatrix}4&2&2&2&1&1\\2&4&2&1&2&1\\2&2&4&1&1&2\\2&1&1&4&2&2\\1&2&1&2&4&2\\1&1&2&2&2&4\\\end{pmatrix}}}$

from which we can recover the λ and r values.

The parameters of a PBIBD(m) satisfy:[28]

1. ${\displaystyle vr=bk}$
2. ${\displaystyle \sum _{i=1}^{m}n_{i}=v-1}$
3. ${\displaystyle \sum _{i=1}^{m}n_{i}\lambda _{i}=r(k-1)}$
4. ${\displaystyle \sum _{u=0}^{m}p_{ju}^{h}=n_{j}}$
5. ${\displaystyle n_{i}p_{jh}^{i}=n_{j}p_{ih}^{j}}$

A PBIBD(1) is a BIBD and a PBIBD(2) in which λ₁  =  λ₂ is a BIBD.[29]

PBIBD(2)s have been studied the most since they are the simplest and most useful of the PBIBDs.[30] They fall into six types[31] based on a classification of the then known PBIBD(2)s by Bose & Shimamoto (1952):[32]

1. group divisible;
2. triangular;
3. Latin square type;
4. cyclic;
5. partial geometry type;
6. miscellaneous.

Suppose that skin cancer researchers want to test three different sunscreens. 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).

A corresponding BIBD can be generated by the R-function design.bib of the R-package agricolae and is specified in the following table:

The investigator chooses the parameters v = 3, k = 2 and λ = 1 for the block design which are then inserted into the R-function. Subsequently, the remaining parameters b and r are determined automatically.

Using the basic relations we calculate that we need b = 3 blocks, that is, 3 test people in order to obtain a balanced incomplete block design. Labeling the blocks A, B and C, to avoid confusion, we have the block design,

A = {2, 3},    B = {1, 3} and C = {1, 2}.

A corresponding incidence matrix is specified in the following table:

Each treatment occurs in 2 blocks, so r = 2.

Just one block (C) contains the treatments 1 and 2 simultaneously and the same applies to the pairs of treatments (1,3) and (2,3). Therefore, λ = 1.

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.