TITLE: Permutation matrices with large distance
QUESTION [5 upvotes]: We say that an order $n$ permutation matrix $P$ has distance $d$ if the Hamming distance between any two $1$-elements of $P$ is at least $d$. For example, the following matrix has distance 2:
$$
\begin{matrix}
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0
\end{matrix}
$$
I am interested in the following question: For which values of $d$ and $n$ do there exist such matrices? More generally, what is the (approximate) number of order $n$ permutation matrices of distance $d$?
I am absolutely sure that these objects have been studied before - there are, after all, a natural variation of error correcting codes. But under what name? Are the answers to my questions already known? I would appreciate any reference. Thank you!

REPLY [7 votes]: I assume that the interpretation with $\ell_\infty$-distance is correct, I.e., the distance between entries $(i,j)$ and $(i’,j’)$ is $\max\{|i-i’|,|j-j’|\}$.
If $n=k^2$, you may achieve $d=k$ but not more.
To achieve $d=k$, enumerate rows and columns from $1$ through $n$ and put ones in all cells of the form $(i, ik\mod (n+1))$, where $i=1,2,\dots,n$.
To see this is optimal, assume that $d=k+1$ is achieved. Consider ones in the first $d$ columns; they must be spread vertically at distances at least $d$ from each other, so we should have $n-1\geq d(d-1)$ which is wrong.