Even before the search for the weight 12 codeword was finished, we were
looking for a way to solve the whole problem. One possibility is to
determine and then the weight enumerator. Unless the weight
enumerator provides a contradiction, we still have to perform another
search. Hall in  proved that if ,
then there exist primitive weight 20 codewords,
which intersect the lines of the plane in at most four points.
We might be able to skip
the step of finding if we could go directly to weight 20.
Unfortunately, we obtained an estimate of at least weight 20
starting configurations. Even though, as Hall remarked, it required only a
few minutes of computer time to try each starting configuration, a few
minutes times equaled years --- clearly beyond our
capability. There was no short cut which could avoid computing the weight
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