A Note on the Generalisation of the Guruswami-Sudan List Decoding Algorithm to Reed-Muller Codes

We revisit the generalisation of the Guruswami–Sudan list decoding algorithm to Reed–Muller codes. Although the generalisation is straightforward, the analysis is more difficult than in the Reed–Solomon case. A previous analysis has been done by Pellikaan and Wu (List decoding of q-ary Reed–Muller codes, Tech. report, from the authors, 2004a; IEEE Trans. on Inf. Th. 50(4): 679–682, 2004b), relying on the theory of Grobner bases We give a stronger form of the well-known Schwartz–Zippel Lemma (Schwartz in J. Assoc. Comput. Mach. 27(4): 701–717, 1980; Zippel in Proc. of EUROSAM 1979, LNCS, vol. 72, Springer, Berlin, pp. 216–226, 1979), taking multiplicities into account. Using this Lemma, we get an improved decoding radius.