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    Let F=(integers) Z_3, the integers mod 3.
    Let V=F^n be the vector space of all n-tuples of F.
    Let B = {{x, y, z} : x, y, z ? V, and x, y, z are distinct, and x + y + z = 0}.
    Show that (V, B) is a Steiner triple system of order 3^n

    On googling 'Steiner triple system', I imagine you want to show the following:

    (i) Given any two a, b in F^n, the number of sets {x, y, z} with x + y + z = 0 containing a and b is independent of the choice of a and b. (hint: there is exactly one such set for each choice of a and b)
    (ii) Given any a in F^n, it lies in precisely 3^n sets {x, y, z} with x + y + z = 0. (hint: suppose a lies in the set, and count how many choices there are for a second element)
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