An empty interval in the spectrum of small weight codewords in the code from points and k-spaces of PG(n, q)

Let Ck(n, q) be the p-ary linear code defined by the incidence matrix of points and k-spaces in PG(n, q), q = ph, p prime, h ≥ 1. In this pa-per, we show that there are no codewords of weight in] q k+1−1 q−1, 2q k [ in Ck(n, q) \ Cn−k(n, q) ⊥ which implies that there are no codewords with this weight in Ck(n, q) \ Ck(n, q) ⊥ if k ≥ n/2. In particular, for the code Cn−1(n, q) of points and hyperplanes of PG(n, q), we exclude all codewords in Cn−1(n, q) with weight in] q n−1 q−1, 2q n−1[. This latter re-sult implies a sharp bound on the weight of small weight codewords of Cn−1(n, q), a result which was previously only known for general dimen-sion for q prime and q = p2, with p prime, p> 11, and in the case n = 2, for q = p3, p ≥ 7.