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On Minors of Maximal Determinant Matrices
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Richard P. Brent

Mathematical Sciences Institute

Australian National University

Canberra, ACT 0200

Australia

Judy-anne H. Osborn

CARMA

The University of Newcastle

Callaghan, NSW 2308

Australia

**Abstract:**

By an old result of Cohn (1965), a Hadamard matrix of order *n* has no
proper Hadamard submatrix of order *m* > *n*/2.
We generalize this result
to maximal determinant submatrices of Hadamard matrices, and show
that an interval of length ∼ *n*/2 is excluded from the allowable orders.
We make a conjecture regarding a lower bound for sums of squares of
minors of maximal determinant matrices, and give evidence to support
it. We give tables of the values taken by the minors of all maximal
determinant matrices of orders 21 and make some observations on the
data. Finally, we describe the algorithms that were used to compute the
tables.

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(Concerned with sequences
A003432
A003433
A215644
A215645.)

Received August 19 2012;
revised version received March 10 2013.
Published in *Journal of Integer Sequences*, March 10 2013.

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