Permutation decoding of first-order Generalized Reed–Muller codes

In a previous work, we described a variation of the classical permutation decoding algorithm that can be applied to any binary affine-invariant code; in particular, it can be applied to first-order Reed–Muller codes successfully. In this paper, we study how to implement it for the family of first-order Generalized Reed–Muller codes. Then, we give examples which show that we improve the number of errors we can correct in comparison with the known results for this family of codes. Finally, we deal, from a probabilistic point of view, with the problem of determining when the algorithm complexity can be reduced.