Hamming distances from a function to all codewords of a Generalized Reed-Muller code of order one
Résumé
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 ${\mathbb F}_q$
from geometric, analytic and algorithmic points of view.
We determine a linear system of $q^{m+1}$ equations and $q^{m+1}$
unknowns, which has for unique solution the Hamming distances of a function in ${\mathcal F}_{(q,m)}$
to all the affine functions. Moreover,
we introduce a Fourier-like transform which allows us to compute all these distances at a cost $O(mq^m)$
and which would be useful for further problems.