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

Abstract

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^h, p prime, h >= 1. In this pa- per, we show that there are no codewords of weight in the open interval ] 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 par- ticular, 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^n-1/q-1, 2q^n-1[. 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 = p^3, p >= 7 ([4],[5],[7],[8]).