[math-fun] poisoned wine bottles -- perfect union codes continued

M(n) is the maximum size "perfect union code" on n symbols. Describe binary codewords of length N in terms of subsets of N symbols. A "perfect union code" is a set of codewords such that if A, B, C are any distinct codewords in the code, then it never happens that A is contained in the union of B and C. --you can obtain perfect union codes using BIBDs -- balanced incomplete block designs. http://en.wikipedia.org/wiki/Block_design#Definition_of_a_BIBD_.28or_2-desig... Any (v,k,lambda) design will do if lambda*2<k and will show M(v)>#blocks=b. For example, any Steiner triple system S(2,3,n) shows M(n) >= n*(n-1)/6 when n mod 6=1 or 3. Hence M(9) >= 12. M(13) >= 26. The Steiner system S(4,7,23) shows M(23) >= 253.