Neo would get lost in this Matrix

Let \(N\) be the number of \(2015\) by \(2015\) matrices in which each entry is either a \(0\) or a \(1\) and each row and column contains an odd number of \(1\)'s.

What is \((\log_{2} N) \pmod{1000}\)?

Image Credit: Wikimedia Matrix Numbers by Jamie Zawinski
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