On complexity of decoding Reed-Muller codes within their code distance

Recently Gopalan, Klivans, and Zuckerman proved that any binary Reed-Muller (RM) code RM(s, m) can be list-decoded up to its minimum distance d with a polynomial complexity of order n in blocklength n. The GKZ algorithm employs a new upper bound that is substantially tighter for RM codes of fixed order s than the universal Johnson bound, and yields a constant number of codewords in a sphere of radius less than d. In this note, we modify the GKZ algorithm and show that full list decoding up to the code distance d can be performed with a lower complexity order of at most n lns−1 n. We also show that our former algorithm yields the same complexity order n lns−1 n if combined with the new GKZ bound on the list size.