250 V Hadamard’s Determinant Problem
We now consider the relationship between designs and Hadamard’s determinant
problem. By passing fromAtoB=(Jn−At)/2, it may be seen immediately that
equality holds in Proposition 19 if and only if there exists a 2-(n,k,λ)design, where
k=(n−s)/2,λ=(n+ 1 − 2 s)/4ands^2 = 2 n−1.
We now show that with any Hadamard matrixA=(αjk)of ordern= 4 dthere is
associated a 2-( 4 d− 1 , 2 d− 1 ,d− 1 )design. Assume without loss of generality that
all elements in the first row and column ofAare 1. We takeP={ 2 ,...,n}as the set
of points andB={B 2 ,...,Bn}as the set of blocks, whereBk={j∈P:αjk= 1 }.
ThenBkhas cardinality|Bk|=n/ 2 −1fork= 2 ,...,n.Moreover,ifTis any subset
ofPwith|T|=2, then the number of blocks containingTisn/ 4 −1. The argument
may also be reversed to show that any 2-( 4 d− 1 , 2 d− 1 ,d− 1 )design is associated
in this way with a Hadamard matrix of order 4d.
In particular, ford=2, the 2-( 7 , 3 , 1 )design associated with the Hadamard matrix
H 2 ⊗H 2 ⊗H 2 ,where
H 2 =
[
11
1 − 1
]
,
is the projective plane of order 2 (Fano plane) illustrated in Figure 1.
The connection between Hadamard matrices and designs may also be derived by a
matrix argument. If
A=
[
1 en− 1
etn− 1 A ̃]
,
is a Hadamard matrix of ordern= 4 d, normalized so that its first row and column
contain only 1’s, thenB=(Jn− 1 + ̃A)/2 is a matrix of 0’s and 1’s such that
J 4 d− 1 B=( 2 d− 1 )J 4 d− 1 , BBt=dI 4 d− 1 +(d− 1 )J 4 d− 1.The optimal spring balance design of order 4d−1, which is obtained by taking
C=(Jn− 1 −A ̃)/2, is a 2-( 4 d− 1 , 2 d,d)design, since
J 4 d− 1 C= 2 dJ 4 d− 1 , CCt=dI 4 d− 1 +dJ 4 d− 1.The notion of 2-design willnow be generalized. Lett,v,k,λbe positive integers
withv≥k≥t.At-(v,k,λ)design,orsimplyat-design,isapair(P,B), whereP
is a set of cardinalityvandBis a collection of subsets ofP, each of cardinalityk,
such that any subset ofPof cardinalitytis contained in exactlyλelements ofB.The
elements ofPwill be calledpointsand the elements ofBwill be calledblocks.At-
(v,k,λ)design withλ=1 is known as aSteiner system.Theautomorphism groupof
at-design is the group of all permutations of the points which map blocks onto blocks.
Ift=1, then each point is contained in exactlyλblocks and so the number of
blocks isλv/k. Suppose now thatt>1. LetSbe a fixed subset ofPof cardinality
t−1andletλ′be the number of blocks which containS. Consider the number of pairs
(T,B), whereB∈B,S⊆T⊆Band|T|=t. By first fixingBand varyingTwe
see that this number isλ′(k−t+ 1 ). On the other hand, by first fixingTand varying
Bwe see that this number isλ(v−t+ 1 ). Hence