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

**Presenter:** Nicholas Cavenagh, Waikato University

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 A_{2m} be the set of (0, 1)-matrices of dimensions 2m x 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 A_{2m} is precisely m^{2}. In this note when m is a power of two we construct an element of A_{2m} which has no defining set of size less than 2m^{2} − o(m^{2}). Given that it is easy to show any A_{2m} has a defining set of size at most 2m^{2}, this construction is asymptotically optimal. Our construction is based on an array, defined using linear algebra, in which any subarray has asymptotically the same number of 0’s and 1’s.

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