# Higher weight spectra of Veronese codes

## TL;DR

This paper investigates the higher weight spectra of Veronese codes over finite fields, providing methods to determine their weight distributions across all field extensions using algebraic and combinatorial tools.

## Contribution

It introduces a novel approach to compute the weight spectra of Veronese codes via Stanley-Reisner rings and matroid theory, extending understanding of their structural properties.

## Key findings

- Higher weight spectra of Veronese codes are explicitly characterized.
- The methods apply to all extension codes over various field extensions.
- The study links algebraic geometry, combinatorics, and coding theory.

## Abstract

We study q-ary linear codes C obtained from Veronese surfaces over finite fields. We show how one can find the higher weight spectra of these codes, or equivalently, the weight distribution of all extension codes of C over all field extensions of the field with q elements. Our methods will be a study of the Stanley-Reisner rings of a series of matroids associated to each code C