Codes from Hall planes of even order

We show that the binary code C of the projective Hall plane Hq2 of even order q 2 where q = 2 t , for t≥2 has words of weight 2q 2 in its hull that are not the difference of the incidence vectors of two lines of Hq2 ; together with an earlier result for the dual Hall planes of even order, this shows that for all t≥2 the Hall plane and its dual are not tame. We also deduce that dim(C)>32t+1, the dimension of the binary code of the desarguesian projective plane of order 22t , thus supporting the Hamada–Sachar conjecture for this infinite class of planes.