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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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