ILL 1986 Problem 46

We wish to construct a matrix with $19$ rows and $86$ columns, with entries $x_{ij} \in \{0, 1, 2\} \ (1 \leq i \leq 19, 1 \leq j \leq 86)$ , such that:

(i) in each column there are exactly $k$ terms equal to $0$ ;

(ii) for any distinct $j, k \in \{1, . . . , 86\}$ there is $i \in \{1, . . . , 19\}$ with $x_{ij} + x_{ik} = 3.$ For what values of $k$ is this possible?