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 = p, p prime, h ≥ 1. In this paper, we show that there are no codewords of weight in the open interval ] q −1 q−1 , 2q[ 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 the open interval ] q −1 q−1 , 2q[. This latter result 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 dimension for q prime and q = p, with p prime, p > 11, and in the case n = 2, for q = p, p ≥ 7 ([4],[5],[7],[8]).