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

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

2012-01-16

arxiv.org/abs/1201.3291v1

Finite Fields Appl. 14 (2008), no. 4, 1020--1038

Mathematics

Combinatorics

Scientific paper

In this paper, we study the p-ary linear code Ck(n, q), q = ph, 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 Ck(n, q)\Ck(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, we present counterexamples to lemmas valid for k >= n/2. Next, we study the dual code of Ck(n, q) and present a lower bound on the weight of the codewords, hence extending the results of Sachar [12] to general dimension.

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