Designs and their codes
Small blocking sets in higher dimensions
Journal of Combinatorial Theory Series A
Small weight codewords in the codes arising from Desarguesian projective planes
Designs, Codes and Cryptography
On the code generated by the incidence matrix of points and hyperplanes in PG(n,q) and its dual
Designs, Codes and Cryptography
On small blocking sets and their linearity
Journal of Combinatorial Theory Series A
On the code generated by the incidence matrix of points and k-spaces in PG(n,q) and its dual
Finite Fields and Their Applications
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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, p11, 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].