Codes from Hall planes of even order

Abstract

We show that the binary code C of the projective Hall plane \({\mathcal{H}_{q^2}}\) of even order q 2 where q = 2t, for \({t \geq 2}\) has words of weight 2q 2 in its hull that are not the difference of the incidence vectors of two lines of \({\mathcal{H}_{q^2}}\) ; together with an earlier result for the dual Hall planes of even order, this shows that for all \({t \geq 2}\) the Hall plane and its dual are not tame. We also deduce that \({{\rm dim}(C) > 3^{2t} + 1}\), 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.