# On the generalized Hamming weights of certain Reed-Muller-type codes

## TL;DR

This paper provides explicit formulas for generalized Hamming weights of affine cartesian and Veronese type codes, extending known combinatorial results to new classes of algebraic codes and their duals.

## Contribution

It introduces an easy-to-evaluate formula for generalized Hamming weights of affine cartesian codes and characterizes parameters of Veronese type codes and their duals.

## Key findings

- Derived a combinatorial formula for affine cartesian codes
- Computed parameters and weights of Veronese type codes
- Linked properties of codes to their duals

## Abstract

There is a nice combinatorial formula of P. Beelen and M. Datta for the $r$-th generalized Hamming weight of an affine cartesian code. Using this combinatorial formula we give an easy to evaluate formula to compute the $r$-th generalized Hamming weight for a family of affine cartesian codes. If $\mathbb{X}$ is a set of projective points over a finite field we determine the basic parameters and the generalized Hamming weights of the Veronese type codes on $\mathbb{X}$ and their dual codes in terms of the basic parameters and the generalized Hamming weights of the corresponding projective Reed--Muller-type codes on $\mathbb{X}$ and their dual codes.