On Minors of Maximal Determinant Matrices
Richard P. Brent
Mathematical Sciences Institute
Australian National University
Canberra, ACT 0200
Judy-anne H. Osborn
The University of Newcastle
Callaghan, NSW 2308
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
Full version: pdf,
(Concerned with sequences
Received August 19 2012;
revised version received March 10 2013.
Published in Journal of Integer Sequences, March 10 2013.
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