OFFSET
1,8
COMMENTS
The complementary depth m(A) of a maximal determinant {+1,-1} matrix of order n is the maximum m < n such that a maximal determinant matrix of order m occurs as a proper submatrix of A, or 0 if n = 1. The depth d(A) of A is d(A) := n - m(A). The depth d(n) is the minimum of d(A) over all maximal determinant matrices A of order n.
We calculated the first 21 terms of the sequence by an exhaustive computation of minors of known maximal determinant matrices as of August 2012.
LINKS
R. P. Brent, The Hadamard Maximal Determinant Problem
Richard P. Brent and Judy-anne H. Osborn, On minors of maximal determinant matrices, arXiv:1208.3819, 2012.
EXAMPLE
For n = 11 the depth is 3 because there is a maximal determinant matrix of order 11 that has a maximal determinant submatrix of order 8 = 11-3, but no larger proper maximal determinant submatrices. Note that only one of the three Hadamard equivalence classes of maximal determinant matrices of order 11 gives depth 3; the others give depth 4, but we take the minimum.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Richard P. Brent and Judy-anne Osborn, Aug 18 2012
STATUS
approved