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Hamming distances from a function to all codewords of a generalized Reed-Muller code of order one. (English) Zbl 1386.94099
Summary: 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.
##### MSC:
 94B05 Linear codes, general 11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
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##### References:
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