Minimum weight and dimension formulas for some geometric codes (Q1963165)

Geometric codes are the duals of the codes defined by the designs associated with finite geometries. The latter are generalized Reed-Muller codes but geometric codes are not, in general, and less is known about them. The principal results of the paper under review are: (1) The minimum weight of the dual of the binary code of the design of points and \(r\)-subspaces of \(PG_m(F_q)\) and that of the design of points and \(r\)-flats of \(AG_m(F_q)\) where \(q\) is even, \(1\leq r\leq m\), \(m \geq 2\), is \((q+2)q^{m-r-1}\). This is obtained using geometric constructions in the associated geometries. (2) The dimension of the dual code of a projective geometry design is a polynomial function in the dimension of the geometry.