How one can combine two covering designs?

Question

There is a discussion on a science forum that how can one find small covering designs for lotto system. Namely, in that lotto we take seven numbers from the set $\{1,\ldots,39\}$ and we win if we have at least four correct number.

I found by Art of problem solving that the proof is based on the inequality $$L(39,7,4,7) \leq L(16,7,4,4) + L(23,7,4,4)$$ (see slide 9 in http://web.archive.org/web/20070824014211/http://www.maths.qmul.ac.uk/~pjc/csgnotes/LottoDesigns.pdf ) But how can one combine these two designs into one design? There is an algorithm in http://arxiv.org/pdf/math/9502238.pdf but I was unable see how it works?

Corresponding discussion at Artofproblemsolving: http://www.artofproblemsolving.com/community/c6h393127 — Max Alekseyev, Sep 22 '15 at 03:56

score 3 · Answer 1 · answered Mar 10 '14 at 11:58

3

Just map the points 1...23 to 17...39 and apply the mapping to each block of the (23,7,4,4) covering design. Then this revised set of blocks plus the set of blocks of the (16,7,4,4) covering design make the desired (39,7,4,7) covering design.

answered Mar 10 '14 at 11:58

Marco

31

NP2P · Answer 2 · 2016-09-26T18:16:29.420

1

You can also download a simple program that will do this job for you. The link is here

edited Sep 26 '16 at 18:16

answered Feb 16 '16 at 10:56

NP2P

9

How one can combine two covering designs?

2 Answers2