Consider an agricultural experiment. Suppose it is desired to compare the yield of v different varieties of grain. It is quite possible that there would be an interaction between the environment (type of soil, rainfall, drainage, etc.) and the variety of grain which would alter the yields. So, b blocks (sets of experimental plots) are chosen in which the environment is fairly consistent throughout the block. In other types of experiments, in which the environment might not be a factor, blocks could be distinguished as plots which receive a particular treatment (say, are given a particular type of fertilizer). In this way, the classification of the experimental plots into blocks and varieties can be used whenever there are two factors which may influence yield.
The obvious technique of growing every variety in a plot in every block, may, for large experiments be too costly or impractical. To deal with this, one would use smaller blocks which did not contain all of the varieties. Now the problem is one of comparison, to minimize the effects of chance due to incomplete blocks, we would want to design the blocks so that the probability of two varieties being compared (i.e. are in the same block) is the same for all pairs. This property would be called balance in the design. Statistical techniques, in particular Analysis of Variance, could then be used to reach conclusions about the experiment.
An example of a (7,7,3,3,1)design is given by the set X consisting of the varieties, 1,2,3,4,5,6,7 and the following blocks:
A (4,4,3,3,2)design is given by X = {1,2,3,4} with blocks:
A slightly larger example is the (8,14,7,4,3)design on the set X = {1,2,3,4,5,6,7,8} with blocks:
Some elementary counting arguments show that there are necessary conditions that the parameters of a BIBD must satisfy.
Theorem VII.1.1  Given a (v,b,r,k,)design,
bk = vr 
r(k1) = (v1). 
Proof: Consider the set of pairs (x,B), where x is a variety and B is a block containing x. By counting this set in two ways we arrive at the first equation. There are v possible values for x, and since each appears in r blocks, vr will count the number of these pairs. On the other hand, there are b blocks and each contains k varieties, so bk also counts the number of these pairs.
The second equation is also obtained by counting. Fix a particular variety, say p, and count the number of pairs of varieties {p,y} where p and y appear in some block together and if the pair appears more than once it is multiply counted. There are v  1 possible choices of y and each such pair will appear in blocks together, so there are (v1) such pairs. On the other hand p appears in r blocks and can be paired with k  1 other elements in such a block, thus r(k1) =(v1).
These conditions on the parameters imply that r and b can be calculated if v,k and are known.
Given a (v,b,r,k,)design, we can represent it with a v × b matrix called the incidence matrix of the design. The rows are labeled with the varieties of the design and the columns with the blocks. We put a 1 in the (i,j)th cell of the matrix if variety i is contained in block j and 0 otherwise. Each row of the incidence matrix has r 1's, each column has k 1's and each pair of distinct rows has common 1's. These observations lead to a useful matrix identity.
Theorem VII.1.2  If A is the incidence matrix of a (v,b,r,k,)design, then
Proof: In the product, an off diagonal entry is the inner product of two distinct rows of A, which must be. A diagonal entry is the inner product of a row of A with itself, and so equals r.
We can use this result to prove another restriction on the parameters of a BIBD, due to Fisher.
Theorem VII.1.3  [Fisher's Inequality] In a (v,b,r,k,)design, b v.
Proof: Suppose that b < v and let A be the incidence matrix of the design. We can add v  b columns of 0's to A to get a v × v matrix B. Since these extra columns of 0's will not alter the inner products, we must have AA^{t} = BB^{t}. By taking determinants we see that,


We can evaluate this determinant by subtracting the first column from each of the other columns and then adding each row to the first row to obtain the following:


So we have that,
If in a BIBD we have v = b (and thus r = k), we say that the BIBD is symmetric or square (maybe, projective). Symmetric BIBD's are often referred to as (v,k,)designs. For symmetric BIBD's, there is an additional constraint on the parameters.
Theorem VII.1.4  [BruckRyserChowla Theorem] The following conditions are necessary for the existence of a symmetric BIBD:
All of the results on the parameters of a BIBD that we have given have been necessary but not sufficient. That is, we can use them to rule out the existence of a BIBD for certain sets of parameters, but given a set of the parameters which satisfy all these conditions does not mean that there actually exists a BIBD with those parameters. There are many sets of possible parameters for which the existence question has not been settled.
Theorem VII.2.2.1  [Kirkman, 1847] There exists a Steiner Triple System of v varieties iff v3 and v1 or 3 mod 6.
Proof: We have already shown the necessity of these conditions. The sufficiency is proved by construction, but as this is rather long and messy we will not present it [details may be found in M. Hall's, Combinatorial Theory].
In 1853 J. Steiner posed the sufficiency of this theorem as a problem and it was proved in 1859 by M. Reiss. Neither mathematician was aware of the fact that the problem had been posed and solved by T.P. Kirkman in an 1847 article appearing in the Cambridge and Dublin Mathematical Journal. Indeed, in 1850 Kirkman went on to pose a more difficult but related problem. This problem, which appeared in "The Lady's and Gentleman's Diary" of 1850, has become to be known as Kirkman's Schoolgirl Problem and was presented as follows:
A teacher would like to take 15 schoolgirls out for a walk, the girls being arranged in 5 rows of three. The teacher would like to ensure equal chances of friendship between any two girls. Hence it is desirable to find different row arrangements for the 7 days of the week such that any pair of girls walk in the same row exactly one day of the week.This problem, which in general asks for a Steiner Triple System on 6t +3 varieties whose blocks can be partitioned into 3t + 1 sets so that any variety appears only once in a set, had not been settled in general until 1971 (RayChaudhuri and Wilson).
There are a number of constructions of STS's, we present only one of them.
Theorem VII.2.2.2  If there is a Steiner Triple System on e varieties and a Steiner Triple System on f varieties then there is a Steiner Triple System on ef varieties.
Proof: Let A be the STS whose elements are a_{1} ,a_{2} ,...,a_{e} and B the STS whose elements are b_{1} ,b_{2} ,...,b_{f} . Let S consist of the ef elements c_{ij}, where i = 1, ...,e and j = 1,...,f. A triple {c_{ir},c_{js},c_{kt}} is in the STS based on S if and only if one of the following conditions hold:
It has been shown that there is only one STS of orders v = 3, 7 or 9. There are two nonisomorphic designs for v = 13 and 80 distinct STS's with v = 15. This work was done by hand but the v = 15 case was verified by computer. Some upper and lower bounds are known for other STS's but no other exact counts.
Of course, it is well known that Steiner Triple Systems are equivalent to idempotent totally symmetric quasigroups and so are connected to Latin Squares.
Let H be an Hadamard matrix of order 4t. First normalize the matrix H (so that the first row and column are just +1's), then remove the first row and column. The 4t1 × 4t1 matrix which remains, say A, has 2t 1's in each row and column and 2t1 +1's in each row and column, so the row and column sums are always 1 for A. The inner product of two distinct rows of A will be 1 and the product of a row with itself will be 4t1. These statements are summarized by the matrix equations,
For example, let H be the 8 × 8 Hadamard matrix in the Hadamard Lecture Notes. In normalized form we have,
+ + + + + + + + + +   + +   +   + +   1 0 0 1 1 0 0 +  +  +  +   +  +  +  0 1 0 1 0 1 0 H = +   + +   +   + +   + 0 0 1 1 0 0 1 + + + +     A = + + +     B = 1 1 1 0 0 0 0 + +     + + +     + + 1 0 0 0 0 1 1 +  +   +  +  +   +  + 0 1 0 0 1 0 1 +   +  + +    +  + +  0 0 1 0 1 1 0So, labeling the rows of B with {1,2,...,7} we have the (7,3,1)design whose 7 blocks are:
Exercise: Prove that an Hadamard design (i.e. a symmetric BIBD with either of these sets of parameters) can be used to construct an Hadamard matrix.
Let L_{1}, L_{2}, ..., L_{q1} be a complete set of MOLS of order q and A a q × q matrix containing q^{2} distinct symbols. We define blocks of size q on this set of q^{2} symbols in the following way: There are q blocks which are the rows of A, and q blocks which are the columns of A. The remaining q^{2}  q blocks are formed by taking each L_{i} in turn, superimposing it on A and taking as blocks the elements of A which correspond to a single symbol in the L_{i}.
As an example consider the set of 3 MOLS of order 4:
1 2 3 4 1 2 3 4 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 3 4 1 2 4 3 2 1 2 1 4 3 4 3 2 1 2 1 4 3 3 4 1 2Now, let A be the matrix,
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16The blocks of the design are:
Exercise: Construct two designs from the 2 MOLS of order 3.
We will first look at three ways to construct new BIBD's from a given symmetric BIBD. The restriction to symmetric BIBD's is a consequence of the following theorem.
Theorem VII.3.1  In a (v,k,)  design, any two blocks have exactly elements in common.
Proof: Let A be the incidence matrix of the design. By Thm. VII.1.2 we have that
Given a (v,k,)design, D, we can form its dual design by interchanging the roles of the varieties and the blocks, that is, we number the blocks of D in any manner and the set of varieties of the dual design is the collection of numbers used, the blocks of the dual are associated to the varieties of D in the following manner, for each variety of D we form a block of the dual design by using all the numbers corresponding to the blocks of D which contain this variety. Theorem VII.3.1 then tells us that every pair of varieties in the dual design is contained in the same number of blocks of the dual design. It is clear that if A is the incidence matrix of the design D, then A^{t} is the incidence matrix of the dual design. It is also not hard to see that the dual of the dual design is the original design. The parameters of the dual design are the same as those of the original. It may turn out that the dual of a design is the original design, in which case we say that the design is selfdual. Consider the (4,3,2) design at the beginning of this chapter, for clarity we will label the blocks with letters, so
Homework: What should the concept of isomorphism mean for block designs? Using your definition, show that the (4,3,2)design above is isomorphic to its dual design.
Another design that can be obtained from a (v,k,)design is the residual design. Pick a block from the original design, remove it and remove all of its varieties from the remaining blocks, the result is the residual design. By theorem VII.3.1, each of the blocks of the original design has varieties in common with the selected block, so the new blocks all have size k . Each variety that remains appears in as many blocks as it did before, that is in r = k blocks. Also, any pair of remaining varieties will still appear together in blocks. The new design then has v  k varieties and only one fewer block, so it is a (vk, v1, k, k,)  design.
Consider the following (11,5,2)design based on {0,1,2,...,10};
Exercise: Given a design whose parameters are the parameters of a residual design, construct a symmetric design having the given design as one of its residual designs.
On the basis of this exercise, knowing that there is no (43,7,1)design implies that there is no (36,42,7,6,1)design.
Yet another design that is obtained from a (v,k,)design is the derived design. To construct this design, we again select a block and remove it, and remove from all the remaining blocks all varieties which are not in the selected block. This will leave us with a (k,v1,k1,,1) design. In the previous example, again selecting the first block we obtain the derived design;
A t(v,k,) design is an ordered pair (S,B), where S is a set of cardinality v, and B is a family of ksubsets (called blocks) of S with the property that each tsubset of S is contained in precisely blocks of B. For nondegeneracy, we shall always assume that 0 < t < k < v. Clearly, a BIBD is a 2(v,k,) design. While not much is known about general tdesigns much work has been done in certain special cases. For each triple satisfying 0 < t < k < v there are many t designs that may be obtained trivially as follows. Let S be any vset. Form C, the set of all possible ksubsets of S. The result (S,C) is a tdesign that we call the full combinatorial design. In this design equals the binomial coefficient C(vt,kt). Let B be the family of ksubsets of S that includes each member of C exactly n times. (S,B) is a t(v,k,n) design. In fact, given any t(v,k,) design we can obtain a t(v,k,n) design by replicating each block n times.
Nontrivial tdesigns have been known for some time for each t, 0 < t < 6. However, it is only recently that it has been shown that tdesigns other than the full combinatorial designs and their replications also exist for all t 6.
An example of a 3(8,4,1) design on the set S = {1,2,...,8} is given by the blocks:
Theorem VII.4.1  Every t(v,k,) design is a (t1)(v,k,*) design where * = (v  t + 1)/(k  t + 1).
Proof: Let D = (S,B) be a t(v,k,) design. Let X be any (t1)subset of S, and L(X) the number of blocks of D that contain X. Clearly X is contained in v  t + 1 tsubsets of S. Let C(X) denote the collection of such subsets. Each of these tsubsets occurs in blocks of D, these being the L(X) blocks that contain X. Now each of these blocks contains k  t + 1 members of C(X). Thus, L(X)(kt+1) = (vt+1). Since L(X) does not depend on the particular (t1)subset chosen, the result follows with * = L(X).
Thus, our example of a 3(8,4,1) design is also a 2(8,4,3) design.
Corollary VII.4.2  Let _{0} = b the number of blocks and _{1} = r the number of blocks containing a given element, and _{i} the number of blocks containing a given isubset, 0 i t. Then,
Proof: This follows from repeated application of Thm. VII.4.1.
Corollary VII.4.3  Let u be an integer such that 0 u t; then the number of subsets in B intersecting a given tsubset in u elements is independent of the tsubset chosen. Proof: This is equivalent to Cor. VII.4.2.
Corollary VII.4.4  The compliment of a tdesign is a tdesign.
Proof: A straightforward application of the principle of inclusionexclusion shows that the complementary design of a t(v,k,) design is a t(v,vk, *) design, where
Theorem VII.4.5  The existence of a t(v,k,) design implies the existence of a (ti)  (vi, ki, ) design.
Proof: Let C(x) be the set of blocks of D = (S,B) that contain a given element x of S. Every (t1)subset of S{x} occurs with x in exactly blocks of D, these blocks being those of C(x)). Thus, (S  {x},B'), where B' is obtained from the blocks of C(x) by removing x, is the required design for i = 1. Repeated application establishes the required result.
Corollary VII.4.6  The existence of a t(v,k,) design implies the existence of a (t1)(v1,k, *) design, where * =  {the # of blocks containing a given t1subset}.
Proof: This design consists of the blocks other than those containing a given element x, that is, the set of blocks B  C(x) in Thm. VII.4.5.
M. Hall, Combinatorial Theory, Blaisdell, Waltham, Mass, 1967.
P.J.Cameron and J.H.van Lint, Graph Theory, Coding Theory and Block Designs Cambridge University Press, Cambridge, 1975. (Also, see the revised and updated conversion to a text by the same authors, Designs, Graphs, Codes and their Links, Cambridge University Press, 1991.)
I.F. Blake and R.C. Mullin, An Introduction to Algebraic and Combinatorial Coding Theory, Academic Press Inc, New York, 1976.
D. Raghavarao, Constructions and Combinatorial Problems in Design of Experiments, Wiley, New York, 1971.
P. Dembowski, Finite Geometries, SpringerVerlag, Berlin, 1968.
H. Ryser,Combinatorial Mathematics, MAA Carus Monographs #14, 1963.
D. Hughes and F.C. Piper, Design Theory, Cambridge University Press, Cambridge, 1985.
An encyclopedic work is:
Th. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1986.