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Enumerate the points of the Fano plane. Consider a vector space V over GF(2) where =7. Let the codewords be constructed as follows. The th position of the codeword is a 1 if the th point belongs to the codeword (geometrically). Else it is a 0.
We need to show that for any given v, there is only one distinct codeword to which we can map (decode) it. Let us consider the weight of an arbitrary v.