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

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.