A215644 Full spectrum threshold for maximal determinant {+1, -1} matrices: largest order of submatrix for which the full spectrum of absolute determinant values occurs.

1, 1, 2, 2, 3, 4, 6, 4, 6, 6, 7, 6, 7, 7, 7, 8, 8, 8, 9, 8, 10

Offset: 1

Author

Richard P. Brent and Judy-anne Osborn, Aug 18 2012

Keywords

• nonn
• hard
• more

Comments

a(n) is the maximum of m(A) taken over all maximal determinant matrices A of order n, where m(A) is the maximum m such that the full spectrum of possible values (ignoring sign) occurs for the minors of order m of A.

Examples

			For n = 8 we have a(8) = 4 as a Hadamard matrix of order 8 has minors of order 4 with the full spectrum of values {0,8,16} (signs are ignored) but minors of order m > 4 do not have this property.
		

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.
• Index entries for sequences related to maximal determinants

Crossrefs

Cf. A003432, A003433, A013588.

Extensions

We calculated the first 21 terms of the sequence by an exhaustive computation of minors of known maximal determinant matrices as at August 2012.