BY-MO 2018 Problem 98

A positive integer $n$ is fixed. Numbers $0$ and $1$ are placed in all cells (exactly one number in any cell) of a $k \times n$ table ( $k$ is a number of the rows in the table, $n$ is the number of the columns in it). We call a table nice if the following property is fulfilled: for any partition of the set of the rows of the table into two nonempty subsets $R$ 1 and $R$ 2 there exists a nonempty set $S$ of the columns such that on the intersection of any row from $R$ 1 with the columns from $S$ there are even number of $1's$ while on the intersection of any row from $R$ 2 with the columns from $S$ there are odd number of $1's$ . Find the greatest number of $k$ such that there exists at least one nice $k \times n$ table.