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

In this paper, we study the p-ary linear code C(PG(n, q)), q = p, p prime, h ≥ 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n, q), and its dual code. We link the codewords of small weight of this code to blocking sets with respect to lines in PG(n, q) and we exclude all possible codewords arising from small linear blocking sets. We also look at the dual code of C(PG(n, q)) and we prove that finding the minimum weight of the dual code can be reduced to finding the minimum weight of the dual code of points and lines in PG(2, q). We present an improved upper bound on this minimum weight and we show that we can drop the divisibility condition on the weight of the codewords in Sachar’s lower bound [12].