Quadratic residue codes, rank three groups and PBIBDs

The automorphism group of the Zetterberg code Z length 17 (also a quadratic residue code) is rank three whose orbits on coordinate pairs determine two strongly regular graphs equivalent to Paley graph attached prime 17. As consequence, codewords given weight are characteristic vectors blocks PBIBD with associate classes cyclic type. More generally, this construction PBIBDs extended codes $$\equiv 1 \pmod {8},$$ adjacency triangular and lattice graphs, various graphs. A remarkable fact existence 2-designs held by 41 for weights 9 10.