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

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.