# Linear codes of 2-designs associated with subcodes of the ternary generalized Reed-Muller codes

## TL;DR

This paper investigates the properties of 2-designs derived from subcodes of ternary generalized Reed-Muller codes, focusing on their incidence matrices, 3-rank, and minimum distance bounds.

## Contribution

It computes the 3-rank of incidence matrices, establishes a lower bound on minimum distance, and proves these codes are subcodes of the 4th order generalized Reed-Muller codes.

## Key findings

- Computed the 3-rank of incidence matrices for these 2-designs.
- Derived a lower bound on the minimum distance of the associated codes.
- Proved these codes are subcodes of the 4th order generalized Reed-Muller codes.

## Abstract

In this paper, the 3-rank of the incidence matrices of 2-designs supported by the minimum weight codewords in a family of ternary linear codes considered in [C. Ding, C. Li, Infinite families of 2-designs and 3-designs from linear codes, Discrete Mathematics 340(10) (2017) 2415--2431] are computed. A lower bound on the minimum distance of the ternary codes spanned by the incidence matrices of these designs is derived, and it is proved that the codes are subcodes of the 4th order generalized Reed-Muller codes.