Number Theory: An Introduction to Mathematics

248 V Hadamard’s Determinant Problem

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Fig. 1.The Fano plane.

Av×bmatrixA=(αij)of 0’s and 1’s is the incidence matrix of a 2-design if
and only if, for some positive integersk,r,λ,

∑v

i= 1

αij=k,

∑b

k= 1

α^2 ik=r,

∑b

k= 1

αikαjk=λ ifi=j( 1 ≤i,j≤v),

or in other words,

evA=keb, AAt=(r−λ)Iv+λJv, (5)

whereenis the 1×nmatrix with all entries 1,Inis then×nunit matrix andJnis the
n×nmatrix with all entries 1.
Designs have been used extensively in the design of agricultural and other experi-
ments. To compare the yield ofvvarieties of a crop onbblocks of land, it would be
expensive to test each variety separately on each block. Instead we can divide each
block intokplots and use a 2-(v,k,λ)design, whereλ=bk(k− 1 )/v(v− 1 ).Then
each variety is used exactlyr=bk/vtimes, no variety is used more than once in any
block, and any two varieties are used together in exactlyλblocks. As an example, take
v=4,b=6,k=2 and henceλ=1,r=3.
Some examples of 2-designs are the finite projective planes. In fact aprojective
planeoforder nmay be defined as a 2-(v,k,λ)design with

v=n^2 +n+ 1 , k=n+ 1 ,λ= 1.

It follows thatb=vandr=k. The blocks in this case are called ‘lines’. The projec-
tive plane of order 2, orFano plane, is illustrated in Figure 1. There are seven points
and seven blocks, the blocks being the six triples of collinear points and the triple of
points on the circle.
Consider now an arbitrary 2-(v,k,λ)design. By (5) and Lemma 15,

det(AAt)=(r−λ)v−^1 (r−λ+λv) > 0 ,

sincer>λ. This implies the inequalityb≥v, due to Fisher (1940), sinceAAtwould
be singular ifb<v.