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

TL;DR

This paper investigates the properties of linear codes derived from incidence matrices in projective geometries, establishing bounds, characterizations, and counterexamples for codewords related to k-blocking sets and their duals.

Contribution

It introduces new bounds and characterizations for codewords in codes from projective geometries, especially relating to minimal k-blocking sets and dual code properties.

Findings

Codewords of certain weights are linked to k-blocking sets.

Minimal k-blocking sets are uniquely reducible.

Counterexamples are provided for specific parameter ranges.

Abstract

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.