| Let S = {0,1,..,16}. Pick k from S.
My A[i] are those subsets of S which have cardinality 5 (a.k.a.
"5-sets"), and the sum of whose elements is k mod 17.
For then suppose that B lies in A[i] ^ A[j], and B has cardinality
4. Then the 5th element of A[i] is determined, and is the same as the
5th element of A[j]. So A[i] = A[j].
How many A[i] are there? Just 1/17 of all 5-sets drawn from S. For the
distribution of the sum of elements over the 5-sets must be fixed under
any perm of S. In particular under the perm (0 1 ... 16) which
increases the sum of any 5-set by 5. Since 5 and 17 are coprime, the
number of possible A[i] is independent of k. This turns out to be 364.
Is 365 possible?
nomadic Andrew in Taipei.
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