# Stability of k mod p multisets and small weight codewords of the code generated by the lines of PG(2, q)

## TL;DR

This paper establishes a stability result for k mod p multisets in PG(2,q), extending understanding of small weight codewords in codes generated by lines, with results valid up to cq sqrt(q).

## Contribution

It provides a new stability theorem for multisets in PG(2,q) and characterizes small weight codewords beyond previous weight bounds.

## Key findings

- Stability result valid up to cq sqrt(q)
- Characterization of non-linear combination codewords by De Boeck and Vandendriessche
- Sharpness of results when q is a square with q > 27 and h >= 4

## Abstract

In this paper, we prove a stability result on k mod p multisets of points in PG(2,q), q = p^h. The particular case k=0 is used to describe small weight codewords of the code generated by the lines of PG(2, q), as linear combination of few lines. Earlier results proved this for codewords with weight less than 2.5q, while our result is valid until cq sqrt(q). It is sharp when 27<q square and h>=4. When q is a prime, De Boeck and Vandendriessche constructed a codeword of weight 3p-3 that is not the linear combination of three lines. We characterise their example.