Codes arising from incidence matrices of points and hyperplanes in $PG(n,q)$

TL;DR

This paper characterizes the second minimum weight codewords in a linear code derived from the incidence matrix of points and hyperplanes in projective space, showing they are scalar multiples of differences of hyperplane incidence vectors.

Contribution

It provides a complete characterization of the second minimum weight codewords in the incidence-based linear code for projective geometries over finite fields.

Findings

Second minimum weight codewords are scalar multiples of differences of hyperplane incidence vectors.

The characterization applies to codes over finite fields with prime power order.

The result enhances understanding of the code's weight distribution and structure.

Abstract

In this paper we completely characterize the words with second minimum weight in the $p-$ary linear code generated by the rows of the incidence matrix of points and hyperplanes of $PG(n,q)$, with $q=p^h$ and $p$ prime, proving that they are the scalar multiples of the difference of the incidence vectors of two distinct hyperplanes of $PG(n,q)$.