On the code generated by the incidence matrix of points and k-spaces in PG(n,q) and its dual

Authors:
M. Lavrauw;L. Storme;G. Van De Voorde
Affiliations:
Ghent University, Department of Pure Mathematics and Computer Algebra, Krijgslaan 281-S22, 9000 Ghent, Belgium;Ghent University, Department of Pure Mathematics and Computer Algebra, Krijgslaan 281-S22, 9000 Ghent, Belgium;Ghent University, Department of Pure Mathematics and Computer Algebra, Krijgslaan 281-S22, 9000 Ghent, Belgium
Venue:
Finite Fields and Their Applications
Year:
2008

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Abstract

In this paper, we study the p-ary linear code C"k(n,q), q=p^h, p prime, h=1, generated by the incidence matrix of points and k-dimensional spaces in PG(n,q). For k=n/2, we link codewords of C"k(n,q)@?C"k(n,q)^@? of weight smaller than 2q^k to k-blocking sets. We first prove that such a k-blocking set is uniquely reducible to a minimal k-blocking set, and exclude all codewords arising from small linear k-blocking sets. For k=n/2. Next, we study the dual code of C"k(n,q) and present a lower bound on the weight of the codewords, hence extending the results of Sachar [H. Sachar, The F"p span of the incidence matrix of a finite projective plane, Geom. Dedicata 8 (1979) 407-415] to general dimension.