A geometric non-existence proof of an extremal additive code

We use a geometric approach to solve an extremal problem in coding theory. Expressed in geometric language we show the non-existence of a system of 12 lines in PG(8,2) with the property that no hyperplane contains more than 5 of the lines. In coding-theoretic terms this is equivalent with the non-existence of an additive quaternary code of length 12, binary dimension 9 and minimum distance 7.