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

摘要

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 ] frac(q~(k + 1) - 1, q - 1), 2 q~k [ in C_k (n, q) {set minus} C_(n - k) (n, q)~⊥ which implies that there are no codewords with this weight in C_k (n, q) {set minus} 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 ] frac(q~n - 1, q - 1), 2 q~(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].