3 research outputs found
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 showed that for binary Reed-Muller codes of length and
distance , one can correct random errors
in time (which is well beyond the worst-case error
tolerance of ).
In this paper, we consider the problem of `syndrome decoding' Reed-Muller
codes from random errors. More specifically, given the
-bit long syndrome vector of a codeword corrupted in
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
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 Saptharishi et al.
2. The second algorithm is obtained by implementing the steps of the
Berlekamp-Welch-style decoding algorithm of Saptharishi et al. in
sublinear-time. The main new ingredient is an algorithm for solving certain
kinds of systems of polynomial equations.Comment: 24 page