The rank of the incidence matrix of points and $d$-flats in finite geometries

The concept of majority decoding and, more generally, threshold decoding was introduced by Massey (ΊQ. In order to obtain majority decodable codes such as (i) a d-th order Projective Geometry code (whose parity check matrix is the incidence matrix of points and d-flats in PG(ί, p)) and (ii) a d-th order Affine Geometry code (whose parity check matrix is the incidence matrix of points other than the origin and J-flats not passing through the origin in EG(ί, p)\ it is necessary to investigate the rank of the incidence matrix of points and d-flats in PG(ί, p) and in EG(ί, p) over GF(p). An exact formula for the rank of the incidence matrix of points and hyperplanes ((ί — l)-flats) has been obtained by Graham and Mac Williams [_2~] for the case t = 2 and has been independently obtained by Smith [5~] and by Goethals and Delsarte [ΛΓ\ for general t. An exact formula for the rank of the incidence matrix of points and d-flats in a special case n = 1 has been obtained by Smith [5]. For general n, although an upper bound for the rank has been obtained by Smith, an explicit formula for the rank has not yet been obtained.*) The purpose of this paper is to derive an explicit formula for the rank of the incidence matrix of points and d-flats in PG(ί, p) and in EG(ί, p) for the general case, by extending the methods used by Smith. The main results are as follows.