On the code generated by the incidence matrix of points and k-spaces in PG(n, q) and its dual

In this paper, we study the p-ary linear code Ck(n, q), q = ph, p prime, h 1, generated by the incidence matrix of points and k-dimensional spaces in PG(n, q). For k n/2, we link codewords of Ck(n, q) \ Ck(n, q)⊥ of weight smaller than 2qk to k-blocking sets. We first prove that such a k-blocking set is uniquely reducible to a minimal k-blocking set, and exclude all codewords arising from small linear k-blocking sets. For k < n/2, we present counterexamples to lemmas valid for k n/2. Next, we study the dual code of Ck(n, q) and present a lower bound on the weight of the codewords, hence extending the results of Sachar [H. Sachar, The Fp span of the incidence matrix of a finite projective plane, Geom. Dedicata 8 (1979) 407–415] to general dimension. © 2008 Published by Elsevier Inc.