# An extension of Delsarte, Goethals and Mac Williams theorem on minimal weight codewords to a class of Reed-Muller type codes

## TL;DR

This paper extends a classical theorem on minimal weight codewords from Reed-Muller codes to a broader class of Reed-Muller type codes over products of finite fields, and discusses the structure of low weight codewords.

## Contribution

It generalizes the minimal weight codeword characterization to Reed-Muller type codes over product fields and explores their low weight codeword structure.

## Key findings

- Extended Delsarte-Goethals-Mac Williams theorem to new code classes
- Characterized minimal weight codewords in these codes
- Analyzed the structure of low weight codewords in affine and projective cases

## Abstract

In 1970 Delsarte, Goethals and Mac Williams published a seminal paper on generalized Reed-Muller codes where, among many important results, they proved that the minimal weight codewords of these codes are obtained through the evaluation of certain polynomials which are a specific product of linear factors, which they describe. In the present paper we extend this result to a class of Reed-Muller type codes defined on a product of (possibly distinct) finite fields of the same characteristic. The paper also brings an expository section on the study of the structure of low weight codewords, not only for affine Reed-Muller type codes, but also for the projective ones.