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If I want to write an $m\times n$ $0/1$ matrix with only rows or columns distinct, I could just pick $m$ or $n$ distinct natural numbers effectively writing them down as rows or columns in $2$-adic form.

Is there a canonical way to write down an $m\times n$ $0/1$ matrix of rank $r$ such that every row is distinct and every column is distinct? Case $m=n$ is most interesting.

Easy criterion is every row and every column has some distinct element which is actually necessary (since we have $0/1$ entries) and sufficient.

If not, what are some tricks and strategies to obtain such a matrix of rank $r$?

If I want to write an $m\times n$ $0/1$ matrix with only rows or columns distinct, I could just pick $m$ or $n$ distinct natural numbers effectively writing them down as rows or columns in base $2$.

Is there a canonical way to write down an $m\times n$ $0/1$ matrix of rank $r$ such that every row is distinct and every column is distinct? Case $m=n$ is most interesting.

If not, what are some tricks and strategies to obtain such a matrix of rank $r$?

If I want to write an $m\times n$ $0/1$ matrix with only rows or columns distinct, I could just pick $m$ or $n$ distinct natural numbers effectively writing them down as rows or columns in $2$-adic form.

Is there a canonical way to write down an $m\times n$ $0/1$ matrix of rank $r$ such that every row is distinct and every column is distinct? Case $m=n$ is most interesting.

If not, what are some tricks and strategies to obtain such a matrix of rank $r$?

If I want to write an $m\times n$ $0/1$ matrix with only rows or columns distinct, I could just pick $m$ or $n$ distinct natural numbers effectively writing them down as rows or columns in $2$-adic form.

Is there a canonical way to write down an $m\times n$ $0/1$ matrix of rank $r$ such that every row is distinct and every column is distinct? Case $m=n$ is most interesting.

Easy criterion is every row and every column has some distinct element which is actually necessary (since we have $0/1$ entries) and sufficient.

If not, what are some tricks and strategies to obtain such a matrix of rank $r$?

I want to write down an $m\times n$ $0/1$ matrix such that every row is distinct and every column is distinct.

If I want to write with only rows or columns distinct, I could just pick $m$ or $n$ distinct natural numbers effectively writing them down as rows or columns.

Is there a canonical way to write down an $m\times n$ $0/1$ matrix such that every row is distinct and every column is distinct? Case $m=n$ is most interesting.

If not what are some tricks and strategies to obtain such a matrix of rank $r$?

If I want to write an $m\times n$ $0/1$ matrix with only rows or columns distinct, I could just pick $m$ or $n$ distinct natural numbers effectively writing them down as rows or columns in $2$-adic form.

Is there a canonical way to write down an $m\times n$ $0/1$ matrix of rank $r$ such that every row is distinct and every column is distinct? Case $m=n$ is most interesting.

If not, what are some tricks and strategies to obtain such a matrix of rank $r$?