# Small weight code words arising from the incidence of points and hyperplanes in PG($\boldsymbol{n,q}$)

## TL;DR

This paper improves bounds on small weight code words in the incidence code of points and hyperplanes in projective space, showing they are generated by hyperplanes through a fixed subspace for large q.

## Contribution

It extends previous results by providing tighter bounds and characterizations of small weight code words in the projective space incidence code for larger q values.

## Key findings

- Code words of weight ≤ (4q - √8q - 33/2)q^{n-2} are linear combinations of hyperplanes through a fixed (n-3)-space.
- For q > 17 and q not in {25,27,29,31,32,49,121}, the structure of small weight code words is fully characterized.
- The bounds on code word weights depend on the value of q, allowing for refined classifications.

## Abstract

Let $C_{n-1}(n,q)$ be the code arising from the incidence of points and hyperplanes in the Desarguesian projective space PG($n,q$). Recently, Polverino and Zullo proved that within this code, all non-zero code words of weight at most $2q^{n-1}$ are scalar multiples of either the incidence vector of one hyperplane, or the difference of the incidence vectors of two distinct hyperplanes. We improve this result, proving that when $q>17$ and $q\notin\{25,27,29,31,32,49,121\}$, all code words of weight at most $(4q-\sqrt{8q}-\frac{33}{2})q^{n-2}$ are linear combinations of incidence vectors of hyperplanes through a fixed $(n-3)$-space. Depending on the omitted value for $q$, we can lower the bound on the weight of $c$ to obtain the same results.