Here it is: lotto649.ws/lotto-wheels/6615-euromillions-wheel.html

@Douglas Zare: Thanks for rewriting the question. It's much clearer now.

Thanks! I wish I could mark more than one answers. Searching in the Internet I found a solution with 36 tuples, and I'll try to find more using your suggestions, but I'm not discarding using a "branch and bound" algorithm.

Thanks a lot! How would you recommend to build the projective 3 space of order 4? I think I'm currently not ready for this task :(.

@alpoge - How can I particularize the concept of 'distance' in your approach of using the domination number? If I understood it correctly, using the domination number to define S3 implies, for each possible tuple, S3 would contain a tuple in which all numbers can differ, but must be directly reachable, i.e. each number with a maximum distance of 1.

@Ashok - I don't think your proposal of $S_3$ such as all members of $S_3$ have a distance $\le$ 3 satisfies my goal. I need $S_3$ as the subset of $S$ that satisfies: - Given any $s$ in $S$, there exists at least one $s_3$ of $S_3$ such as $d_h(s, s_3) \le 3$ I'm sorry for my lack of accuracy. The reason behind the question is to find the set $S_3$ to be sure that, for any possible tuple of $S$, I can find an element in $S_3$ subset matching 2 or more numbers of the given tuple.

Yeah, sorry. $S$ is a 5-element tuple consisting of numbers between 1 and 50. They are ordered and cannot contain duplicates. $d_h$ is the number of differences between two given tuples: 5 - number of matching numbers. For example, to convert (1,2,3,4,5) into (1,3,5,6,7) I'd have to replace 2 with 6 and 4 with 7. So the distance is 2.