On the code generated by the incidence matrix of points and k-spaces in [formula omitted] and its dual

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 2 q 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 , we present counterexamples to lemmas valid 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.