Minimum Weight and Dimension Formulas for Some Geometric Codes

The geometric codes are the duals of the codes defined by the designs associated with finite geometries. The latter are generalized Reed–Muller codes, but the geometric codes are, in general, not. We obtain values for the minimum weight of these codes in the binary case, using geometric constructions in the associated geometries, and the BCH bound from coding theory. Using Hamada's formula, we also show that the dimension of the dual of the code of a projective geometry design is a polynomial function in the dimension of the geometry.