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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 ${\mathbbF}^q$. We introduce a transformation that allows us to determine a linearsystem of $q^{m+1}$ equations and $q^{m+1}$ unknowns, which has for solutionthe Hamming distances of a function in ${\mathcal F}_{(q,m)}$ to all the affinefunctions.

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