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

TL;DR

This paper investigates the properties of linear codes derived from the incidence matrix of points and hyperplanes in projective spaces PG(n, q), linking codewords to blocking sets and improving bounds on dual code weights.

Contribution

It establishes connections between small weight codewords and blocking sets, and provides improved bounds on the minimum weight of the dual code in PG(2, q).

Findings

Small weight codewords relate to blocking sets.

Excluded codewords from small linear blocking sets.

Improved upper bounds on dual code minimum weight.

Abstract

In this paper, we study the p-ary linear code C(PG(n, q)), q = p^h, p prime, h >= 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n, q), and its dual code. We link the codewords of small weight of this code to blocking sets with respect to lines in PG(n, q) and we exclude all possible codewords arising from small linear blocking sets. We also look at the dual code of C(PG(n,q)) and we prove that find- ing the minimum weight of the dual code can be reduced to finding the minimum weight of the dual code of points and lines in PG(2, q). We present an improved upper bound on this minimum weight and we show that we can drop the divisibility condition on the weight of the codewords in Sachar's lower bound [12].