an:06815207
Zbl 1386.94099
Abd??n, Miriam; Rolland, Robert
Hamming distances from a function to all codewords of a generalized Reed-Muller code of order one
EN
Appl. Algebra Eng. Commun. Comput. 28, No. 5, 387-408 (2017).
00371967
2017
j
94B05 11T71
Reed-Muller code; Hamming distance; arrangement of hyperplanes
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.