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I am interested in binary matrices with a near balanced number of matched and mismatched entries between columns. For example, a Hadamard matrix has a perfect balance between matched and mismatched entries between columns: \begin{bmatrix} 1 1 1 1 \\ 1 0 1 0\\ 1 1 0 0\\ 1 0 0 1 \end{bmatrix} Concretely, we see that $[1 1 1 1]$ and $[1 0 1 0]$ match in the first and third and entries and mismatch in the second and fourth entries.

However, we can also construct matrices that have nearly balanced matched and mismatched entries. For example, by deleting the last row of the above we obtain a matrix where the number of matched and mismatched entries between two columns are different by at most one: \begin{bmatrix} 1 1 1 1 \\ 1 0 1 0\\ 1 1 0 0\\ \end{bmatrix} Concretely, we see that $[1 1 1]$ and $[1 0 1]$ match in the first and third entry and are mismatched in the second entry. The number of matches and mismatches differs by one.

Note that this is different than the concept behind error correcting codes, where we try and maximize the minimum number of mismatches between any two columns.

What are these objects with nearly balanced matched and mismatched columns called? If I want to learn more about them, what kind of topics will be relevant?

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