Minimal codewords arising from the incidence of points and hyperplanes in projective spaces

TL;DR

This paper investigates minimal codewords in codes derived from point-hyperplane incidences in projective spaces, showing that small weight codewords can be expressed as combinations of a few hyperplanes, with implications for their minimality.

Contribution

The work establishes a bound on small weight codewords in incidence codes and characterizes their structure as linear combinations of hyperplanes, especially for large, non-prime q.

Findings

Small weight codewords are linear combinations of few hyperplanes.

A bound on the weight of small codewords is provided.

A graph-theoretical condition for minimality of these codewords is derived.

Abstract

Over the past few years, the codes $\mathcal{C}_{n-1}(n,q)$ arising from the incidence of points and hyperplanes in the projective space $\text{PG}(n,q)$ attracted a lot of attention. In particular, small weight codewords of $\mathcal{C}_{n-1}(n,q)$ are a topic of investigation. The main result of this work states that, if $q$ is large enough and not prime, a codeword having weight smaller than roughly $\frac{1}{2^{n-2}}q^{n-1}\sqrt{q}$ can be written as a linear combination of a few hyperplanes. Consequently, we use this result to provide a graph-theoretical sufficient condition for these codewords of small weight to be minimal.