Minimum weight and dimension formulas for some geometric codes
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.
scientific article; zbMATH DE number 1392724