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Hadamard Matrices
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Introduction
Discrepancy and Ramsey Theory for
-Matrices
Lemma 2.1
(Lindsey's Lemma)
Let
be a Hadamard matrix. Let
and
,
. Then
Definition 2.2
We call the submatrix on the entries corresponding to
an
rectangle
in
. We call the sum
the
discrepancy
of this rectangle.
Discrepancy measures the deviation from uniform distribution.
Definition 2.3
A rectangle is
homogeneous
if all of its entries are equal.
O
PEN
P
ROBLEM
2.4
Construct an explicit family of
matrices
(for infinitely many values of
) such that
has no homogeneous
rectangles for
.
Laszlo Babai 2003-06-19