Hamming distances from a function to all codewords of a Generalized Reed-Muller code of order one

TL;DR

This paper introduces a transformation method to compute the Hamming distances from any function over a finite field to all affine functions, aiding the analysis of generalized Reed-Muller codes of order one.

Contribution

The paper presents a novel transformation that simplifies the calculation of Hamming distances to all affine functions for functions over finite fields, advancing coding theory analysis.

Findings

Provides a linear system to determine Hamming distances

Enables efficient analysis of generalized Reed-Muller codes

Offers a new tool for coding theory research

Abstract

For any finite field ${\mathbb F}_q$ with $q$ elements, we study the set ${\mathcal F}_{(q,m)}$ of functions from ${\mathbb F}_q^m$ into ${\mathbb F}^q$. We introduce a transformation that allows us to determine a linear system of $q^{m+1}$ equations and $q^{m+1}$ unknowns, which has for solution the Hamming distances of a function in ${\mathcal F}_{(q,m)}$ to all the affine functions.