Block design

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This article is about block designs with fixed block size (uniform). For block designs with variable block sizes, see Combinatorial design. For experimental designs in statistics, see randomized block design.

In combinatorial mathematics, a block design is a set together with a family of subsets (repeated subsets are allowed at times) whose members are chosen to satisfy some set of properties that are deemed useful for a particular application. These applications come from many areas, including experimental design, finite geometry, software testing, cryptography, and algebraic geometry. Many variations have been examined, but the most intensely studied are the balanced incomplete block designs (BIBDs or 2-designs) which historically were related to statistical issues in the design of experiments.[1][2]

A block design in which all the blocks have the same size is called uniform. The designs discussed in this article are all uniform. Pairwise balanced designs (PBDs) are examples of block designs that are not necessarily uniform.

Definition of a BIBD (or 2-design)

Given a finite set X (of elements called points) and integers k, r, λ ≥ 1, we define a 2-design (or BIBD, standing for balanced incomplete block design) B to be a family of k-element subsets of X, called blocks, such that the number r of blocks containing x in X is not dependent on which x is chosen, and the number λ of blocks containing given distinct points x and y in X is also independent of the choices.

"Family" in the above definition can be replaced by "set" if repeated blocks are not allowed. Designs in which repeated blocks are not allowed are called simple.

Here v (the number of elements of X, called points), b (the number of blocks), k, r, and λ are the parameters of the design. (To avoid degenerate examples, it is also assumed that v > k, so that no block contains all the elements of the set. This is the meaning of "incomplete" in the name of these designs.) In a table:

 v points, number of elements of X b number of blocks r number of blocks containing a given point k number of points in a block λ number of blocks containing any 2 (or more generally t) specific points

The design is called a (v, k, λ)-design or a (v, b, r, k, λ)-design. The parameters are not all independent; v, k, and λ determine b and r, and not all combinations of v, k, and λ are possible. The two basic equations connecting these parameters are

${\displaystyle bk=vr,\,}$
${\displaystyle \lambda (v-1)=r(k-1).\,}$

These conditions are not sufficient as, for example, a (43,7,1)-design does not exist.[3]

The order of a 2-design is defined to be n = r − λ. The complement of a 2-design is obtained by replacing each block with its complement in the point set X. It is also a 2-design and has parameters v′ = v, b′ = b, r′ = b − r, k′ = v − k, λ′ = λ + b − 2r. A 2-design and its complement have the same order.

A fundamental theorem, Fisher's inequality, named after the statistician Ronald Fisher, is that b ≥ v in any 2-design.

Symmetric BIBDs

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.[4] Symmetric designs have the smallest number of blocks amongst all the 2-designs with the same number of points.

In a symmetric design r = k holds as well as b = v, and, while it is generally not true in arbitrary 2-designs, in a symmetric design every two distinct blocks meet in λ points.[5] A theorem of Ryser provides the converse. If X is a v-element set, and B is a v-element set of k-element subsets (the "blocks"), such that any two distinct blocks have exactly λ points in common, then (X, B) is a symmetric block design.[6]

The parameters of a symmetric design satisfy

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

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.

The following are important examples of symmetric 2-designs:

Projective planes

Main article: Projective plane

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.[7]

Biplanes

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.[7] 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[8] 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 and the blocks are the faces of a tetrahedron.
• 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.[9]
• 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.[10]
• There are three biplanes of order 4 (and 16 points, lines of size 6; a 2-(16,6,2)). 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)).[11]
• There are five biplanes of order 9 (and 56 points, lines of size 11; a 2-(56,11,2)).[12]
• Two biplanes are known of order 11 (and 79 points, lines of size 13; a 2-(79,13,2)).[13]

Hadamard 2-designs

An Hadamard matrix of size m is an m × m matrix H whose entries are ±1 such that HH  = mIm, where H is the transpose of H and Im 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.[14] 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.

Resolvable 2-designs

A resolvable 2-design is a BIBD whose blocks can be partitioned into sets (called parallel classes), each of which forms a partition of the point set of the BIBD. The set of parallel classes is called a resolution of the design.

If a 2-(v,k,λ) resolvable design has c parallel classes, then b  ≥ v + c − 1.[15]

Consequently, a symmetric design can not have a non-trivial (more than one parallel class) resolution.[16]

Archetypical resolvable 2-designs are the finite affine planes. A solution of the famous 15 schoolgirl problem is a resolution of a 2-(15,3,1) design.[17]

Generalization: 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

${\displaystyle \lambda _{i}=\lambda \left.{\binom {v-i}{t-i}}\right/{\binom {k-i}{t-i}}{\text{ for }}i=0,1,\ldots ,t,}$

where λi is the number of blocks that contain any i-element set of points.

Theorem:[18] Any t-(v,k,λ)-design is also an s-(v,ks)-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.

There are no known examples of non-trivial t-(v,k,1)-designs with ${\displaystyle t>5}$.

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:[19] 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.[20]

Every Hadamard 2-design is extendable (to an Hadamard 3-design).[21]

Theorem:.[22] 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.[23]

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).

Steiner systems

Main article: Steiner system

A Steiner system (named after Jakob Steiner) is a t-design with λ = 1 and t ≥ 2.

A Steiner system with parameters t, k, n, written S(t,k,n), is an n-element set S together with a set of k-element subsets of S (called blocks) with the property that each t-element subset of S is contained in exactly one block. In the general notation for block designs, an S(t,k,n) would be a t-(n,k,1) design.

This definition is relatively modern, generalizing the classical definition of Steiner systems which in addition required that k = t + 1. An S(2,3,n) was (and still is) called a Steiner triple system, while an S(3,4,n) was called a Steiner quadruple system, and so on. With the generalization of the definition, this naming system is no longer strictly adhered to.

Projective planes and affine planes are examples of Steiner systems under the current definition while only the Fano plane (projective plane of order 2) would have been a Steiner system under the older definition.

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:

• ${\displaystyle R_{0}=\{(x,x):x\in X\}}$ and is called the Identity relation.
• Defining ${\displaystyle R^{*}:=\{(x,y)|(y,x)\in R\}}$, if R in S, then R* in S
• If ${\displaystyle (x,y)\in R_{k}}$, the number of ${\displaystyle z\in X}$ such that ${\displaystyle (x,z)\in R_{i}}$ and ${\displaystyle (z,y)\in R_{j}}$ is a constant ${\displaystyle p_{ij}^{k}}$ depending on i, j, k but not on the particular choice of x and y.

An association scheme is commutative if ${\displaystyle p_{ij}^{k}=p_{ji}^{k}}$ for all i, j and k. Most authors assume this property.

A partially balanced incomplete block design with n associate classes (PBIBD(n)) is a block design based on a v-set X with b blocks each of size k and with each element appearing in r blocks, such that there is an association scheme with n classes defined on X where, if elements x and y are ith associates, 1 ≤ in, then they are together in precisely λi blocks.

A PBIBD(n) determines an association scheme but the converse is false.[24]

Example

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 Rs.

1 2 3 4 5 6
1  0   1   1   2   3   3
2  1   0   1   3   2   3
3  1   1   0   3   3   2
4  2   3   3   0   1   1
5  3   2   3   1   0   1
6  3   3   2   1   1   0

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

 124 134 235 456 125 136 236 456

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.[25]

Properties

The parameters of a PBIBD(m) satisfy:[26]

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.[27]

Two associate class PBIBDs

PBIBD(2)s have been studied the most since they are the simplest and most useful of the PBIBDs.[28] They fall into six types[29] based on a classification of the then known PBIBD(2)s by Bose & Shimamoto (1952):[30]

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

Applications

The mathematical subject of block designs originated in the statistical framework of design of experiments. These designs were especially useful in applications of the technique of analysis of variance (ANOVA). This remains a significant area for the use of block designs.

While the origins of the subject are grounded in biological applications (as is some of the existing terminology), the designs are used in many applications where systematic comparisons are being made, such as in software testing.

The incidence matrix of block designs provide a natural source of interesting block codes that are used as error correcting codes. The rows of their incidence matrices are also used as the symbols in a form of pulse-position modulation.[31]

Notes

1. ^ Colbourn & Dinitz 2007
2. ^ Stinson 2003, pg.1
3. ^ Proved by Tarry in 1900 who showed that there was no pair of orthogonal Latin squares of order six. The 2-design with the indicated parameters is equivalent to the existence of five mutually orthogonal Latin squares of order six.
4. ^ They have also been referred to as projective designs or square designs. These alternatives have been used in an attempt to replace the term "symmetric", since there is nothing symmetric (in the usual meaning of the term) about these designs. The use of projective is due to P.Dembowski (Finite Geometries, Springer, 1968), in analogy with the most common example, projective planes, while square is due to P. Cameron (Designs, Graphs, Codes and their Links, Cambridge, 1991) and captures the implication of v = b on the incidence matrix. Neither term has caught on as a replacement and these designs are still universally referred to as symmetric.
5. ^ Stinson 2003, pg.23, Theorem 2.2
6. ^ Ryser 1963, pp. 102–104
7. ^ a b Hughes & Piper 1985, pg.109
8. ^ Hall 1986, pp.320-335
9. ^ Assmus & Key 1992, pg.55
10. ^ Martin, Pablo; Singerman, David (April 17, 2008), From Biplanes to the Klein quartic and the Buckyball (PDF), p. 4
11. ^ Salwach & Mezzaroba 1978
12. ^ Kaski & Östergård 2008
13. ^ Aschbacher 1971, pp. 279–281
14. ^ Stinson 2003, pg. 74, Theorem 4.5
15. ^ Hughes & Piper 1985, pg. 156, Theorem 5.4
16. ^ Hughes & Piper 1985, pg. 158, Corollary 5.5
17. ^ Beth, Jungnickel & Lenz 1986, pg. 40 Example 5.8
18. ^ Stinson 2003, pg.203, Corollary 9.6
19. ^ Hughes & Piper 1985, pg.29
20. ^ Cameron & van Lint 1991, pg. 11, Proposition 1.34
21. ^ Hughes & Piper 1985, pg. 132, Theorem 4.5
22. ^ Cameron & van Lint 1991, pg. 11, Theorem 1.35
23. ^ Colbourn & Dinitz 2007, pg. 114, Remarks 6.35
24. ^ Street & Street 1987, pg. 237
25. ^ Street & Street 1987, pg. 238
26. ^ Street & Street 1987, pg. 240, Lemma 4
27. ^ Colburn & Dinitz 2007, pg. 562, Remark 42.3 (4)
28. ^ Street & Street 1987, pg. 242
29. ^ Not a mathematical classification since one of the types is a catch-all "and everything else".
30. ^ Raghavarao 1988, pg. 127
31. ^ Noshad, Mohammad; Brandt-Pearce, Maite (Jul 2012). "Expurgated PPM Using Symmetric Balanced Incomplete Block Designs". IEEE Communications Letters. 16 (7): 968–971. doi:10.1109/LCOMM.2012.042512.120457.

References

• Aschbacher, Michael (1971). "On collineation groups of symmetric block designs". Journal of Combinatorial Theory, Series A. 11 (3): 272–281. doi:10.1016/0097-3165(71)90054-9.
• Assmus, E.F.; Key, J.D. (1992), Designs and Their Codes, Cambridge: Cambridge University Press, ISBN 0-521-41361-3
• Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried (1986), Design Theory, Cambridge: Cambridge University Press. 2nd ed. (1999) ISBN 978-0-521-44432-3.
• R. C. Bose, "A Note on Fisher's Inequality for Balanced Incomplete Block Designs", Annals of Mathematical Statistics, 1949, pages 619–620.
• Bose, R. C.; Shimamoto, T. (1952), "Classification and analysis of partially balanced incomplete block designs with two associate classes", Journal of the American Statistical Association, 47: 151–184, doi:10.1080/01621459.1952.10501161
• Cameron, P. J.; van Lint, J. H. (1991), Designs, Graphs, Codes and their Links, Cambridge: Cambridge University Press, ISBN 0-521-42385-6
• Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs (2nd ed.), Boca Raton: Chapman & Hall/ CRC, ISBN 1-58488-506-8
• R. A. Fisher, "An examination of the different possible solutions of a problem in incomplete blocks", Annals of Eugenics, volume 10, 1940, pages 52–75.
• Hall, Jr., Marshall (1986), Combinatorial Theory (2nd ed.), New York: Wiley-Interscience, ISBN 0-471-09138-3
• Hughes, D.R.; Piper, E.C. (1985), Design theory, Cambridge: Cambridge University Press, ISBN 0-521-25754-9
• Kaski, Petteri & Östergård, Patric (2008). "There Are Exactly Five Biplanes with k = 11". Journal of Combinatorial Designs. 16 (2): 117–127. doi:10.1002/jcd.20145. MR 2008m:05038.
• Lander, E. S. (1983), Symmetric Designs: An Algebraic Approach, Cambridge: Cambridge University Press
• Lindner, C.C.; Rodger, C.A. (1997), Design Theory, Boca Raton: CRC Press, ISBN 0-8493-3986-3
• Raghavarao, Damaraju (1988). Constructions and Combinatorial Problems in Design of Experiments (corrected reprint of the 1971 Wiley ed.). New York: Dover.
• Raghavarao, Damaraju & Padgett, L.V. (2005). Block Designs: Analysis, Combinatorics and Applications. World Scientific.
• Ryser, Herbert John (1963), "Chapter 8: Combinatorial Designs", Combinatorial Mathematics (Carus Monograph #14), Mathematical Association of America
• Salwach, Chester J.; Mezzaroba, Joseph A. (1978). "The four biplanes with k = 9". Journal of Combinatorial Theory, Series A. 24 (2): 141–145. doi:10.1016/0097-3165(78)90002-X.
• van Lint, J.H.; Wilson, R.M. (1992). A Course in Combinatorics. Cambridge: Cambridge University Press.