Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.2

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 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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