Syndrome decoding of Reed-Muller codes and tensor decomposition over finite fields

Reed-Muller codes are some of the oldest and most widely studied error-correcting codes, of interest for both their algebraic structure as well as their many algorithmic properties. A recent beautiful result of Saptharishi, Shpilka and Volk [SSV17] showed that for binary Reed-Muller codes of length n and distance d = O (1), one can correct polylog (n) random errors in poly (n) time (which is well beyond the worst-case error tolerance of O (1)). In this paper, we consider the problem of syndrome decoding Reed-Muller codes from random errors. More specifically, given the polylog (n)-bit long syndrome vector of a codeword corrupted in polylog(n) random coordinates, we would like to compute the locations of the codeword corruptions. This problem turns out to be equivalent to a basic question about computing tensor decomposition of random low-rank tensors over finite fields. Our main result is that syndrome decoding of Reed-Muller codes (and the equivalent tensor decomposition problem) can be solved efficiently, i.e., in polylog (n) time. We give two algorithms for this problem: 1. The first algorithm is a finite field variant of a classical algorithm for tensor decomposition over real numbers due to Jennrich. This also gives an alternate proof for the main result of [SSV17]. 2. The second algorithm is obtained by implementing the steps of [SSV17]'s Berlekamp-Welch-style decoding algorithm in sublinear-time. The main new ingredient is an algorithm for solving certain kinds of systems of polynomial equations.