The football pool problem for 5 matches

We consider the set N of all 5-tuples x = (xl, x2, x3, x4, xs) with x~ = 0, 1, or 2 for i = 1,..., 5. The problem treated in this paper is determining the minimal k for which a set ~ of 5-tuples exists such that for each x in N there is an element in ~ that differs from x in at most one coordinate.