MathNet 2012 Question 74

Let $n$ be a positive integer. Determine the largest integer $m$ for which an array with $m$ rows and $n$ columns can be filled with real numbers so that for any two different rows $[a_1,a_2,\ldots,a_n]$ and $[b_1,b_2,\ldots,b_n]$,
$$\max\bigl(|a_1-b_1|,|a_2-b_2|,\ldots,|a_n-b_n|\bigr)=1.$$