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

Let C"k(n,q) be the p-ary linear code defined by the incidence matrix of points and k-spaces in PG(n,q), q=p^h, p prime, h>=1. In this paper, we show that there are no codewords of weight in the open interval ]q^k^+^1-1q-1,2q^k[ in C"k(n,q)@?C"n"-"k(n,q)^@? which implies that there are no codewords with this weight in C"k(n,q)@?C"k(n,q)^@? if k>=n/2. In particular, for the code C"n"-"1(n,q) of points and hyperplanes of PG(n,q), we exclude all codewords in C"n"-"1(n,q) with weight in the open interval ]q^n-1q-1,2q^n^-^1[. This latter result implies a sharp bound on the weight of small weight codewords of C"n"-"1(n,q), a result which was previously only known for general dimension for q prime and q=p^2, with p prime, p>11, and in the case n=2, for q=p^3, p>=7 [K. Chouinard, On weight distributions of codes of planes of order 9, Ars Combin. 63 (2002) 3-13; V. Fack, Sz.L. Fancsali, L. Storme, G. Van de Voorde, J. Winne, Small weight codewords in the codes arising from Desarguesian projective planes, Des. Codes Cryptogr. 46 (2008) 25-43; M. Lavrauw, L. Storme, G. Van de Voorde, On the code generated by the incidence matrix of points and hyperplanes in PG(n,q) and its dual, Des. Codes Cryptogr. 48 (2008) 231-245; M. Lavrauw, L. Storme, G. Van de Voorde, On the code generated by the incidence matrix of points and k-spaces in PG(n,q) and its dual, Finite Fields Appl. 14 (2008) 1020-1038].