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?
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.