Constructing (0,1)-matrices with large minimal defining sets

Abstract

If D is a partially filled-in (0, 1)-matrix with a unique completion to a (0, 1)-matrix M (with prescribed row and column sums), we say that D is a defining set for M. Let A₂ₘ,ₘbe the set of (0, 1)-matrices of dimensions 2m×2m with uniform row and column sum m. It is shown in (Cavenagh, 2013) that the smallest possible size for a defining set of an element of A₂ₘ,ₘ is precisely m². In this note when m is a power of two we construct an element of A₂ₘ,ₘ which has no defining set of size less than 2m² − o(m²). Given that it is easy to show any A₂ₘ,ₘ has a defining set of size at most 2m², this construction is asymptotically optimal. Our construction is based on a array, defined using linear algebra, in which any subarray has asymptotically the same number of 0’s and 1’s.