Abstract | ||
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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.
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Year | DOI | Venue |
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2017 | 10.5555/3174304.3175314 | SODA '18: Symposium on Discrete Algorithms
New Orleans
Louisiana
January, 2018 |
DocType | Volume | ISBN |
Journal | abs/1712.06039 | 978-1-61197-503-1 |
Citations | PageRank | References |
0 | 0.34 | 2 |
Authors | ||
2 |
Name | Order | Citations | PageRank |
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Swastik Kopparty | 1 | 384 | 32.89 |
Aditya Potukuchi | 2 | 0 | 1.01 |