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# Steiner triple system help!!! watch

1. 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
2. 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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Updated: December 16, 2010
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