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 paper, we show that there are no codewords of weight in the open interval 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 . 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=p2, with p prime, p>11, and in the case n=2, for q=p3, 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].