Reed–Muller codes and permutation decoding

We show that the first- and second-order Reed–Muller codes, R ( 1 , m ) and R ( 2 , m ) , can be used for permutation decoding by finding, within the translation group, ( m − 1 ) - and ( m + 1 ) -PD-sets for R ( 1 , m ) for m ≥ 5 , 6 , respectively, and ( m − 3 ) -PD-sets for R ( 2 , m ) for m ≥ 8 . We extend the results of Seneviratne [P. Seneviratne, Partial permutation decoding for the first-order Reed-Muller codes, Discrete Math., 309 (2009), 1967–1970].