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Up: Constructing G (and hence,
Choose an arbitrary codeword c of C. Counting the number of vectors within radius 1, we have c itself and vectors (since we have options for each of the possible errors). We have shown these spheres are mutally disjoint, therefore exactly
vectors of V are covered by the spheres of C. Thus for a linear 1-error correcting code, C is perfect if and only if
, since that's how many vectors are in V.
Thus to show our Hamming code is perfect, we use the fact that the size of C is . Plugging this into our equation from the above theorem we have
, where is an integer.
Which is how is chosen by definition of our Hamming code.